Aeroelastic Concepts for Flexible Aircraft Structures

AEROELASTIC CONCEPTS FOR FLEXIBLE AIRCRAFT STRUCTURES

Sebastian Heinze Aeronautical and Vehicle Engineering Royal Institute of Technology SE-100 44 Stockholm, Sweden

TRITA/AVE 2007:29 ISBN 978-91-7178-706-4 TRITA/AVE 2007:29 KTH Farkost & Flyg ISSN 1651-7660 Teknikringen 8 ISRN KTH/AVE/DA-07:29-SE 100 44 Stockholm ISBN 978-91-7178-706-4

Akademisk avhandling som med tillst˚and av Kungliga Tekniska hogskolan¨ fredagen den 15 juni 2007 kl. 13.15, Kollegiesalen F3, Lindstedtsvägen 26, Stock- holm, for¨ teknisk doktorsgrads vinnande framlägges till offentlig granskning av Sebastian Heinze.

Typsatt i LATEX Tryck: Universitetsservice US AB, Stockholm 2007 c Sebastian Heinze 2007 Aeroelastic Concepts for Flexible Aircraft Structures 1

Preface

The work presented in this doctoral thesis was performed between April 2003 and May 2007 at the Department of Aeronautical and Vehicle Engineering at the Royal Institute of Technology (KTH) in Stockholm, Sweden. Financial support has been provided by the European research project Active Aeroelastic Aircraft Structures (3AS), the Swedish Defence Materiel Administration (FMV) and the Swedish National Program for Aeronautics Research (NFFP). I would like to express my gratitude to my supervisor, Dan Borglund, for outstanding support and advice during the past years, and for all the good times together. Furthermore, many thanks to the head of the Division of Flight Dynamics, Ulf Ringertz, and to my former and current colleagues at Far & Flyg, in particular Carin, David, Fredrik, Gloria, Marianne, Martin, Martin, and Ulrik for all companionship and inspiration. Also, many thanks to Professor Moti Karpel for his contribution to my work. I am very grateful to my family and friends for encouraging and supporting me at all times. Finally, special thanks to Maria for all the love and understand- ing, and for giving me strength in the recent years.

Aeroelastic Concepts for Flexible Aircraft Structures 3

Abstract

In this thesis, aeroelastic concepts for increased aircraft performance are developed and evaluated. Active aeroelastic concepts are in focus as well as robust analysis concepts aiming at efficient analysis using numerical models with uncertain or varying model parameters. The thesis presents different approaches for exploitation of fluid-structure interaction of active aeroelastic structures. First, a high aspect ratio wing in wind tunnel testing conditions is considered. The wing was developed within the European research project Active Aeroelastic Aircraft Structures and used to demonstrate how structural flexibility can be exploited by using multiple control surfaces such that the deformed wing shape gives minimum drag for different flight conditions. Two different drag minimization studies are presented, one aiming at reduced induced drag based on numerical optimization techniques, another one aiming at reduced measured total drag using real-time optimization in the wind tunnel experiment. The same wing is also used for demonstration of an active concept for gust load alleviation using a piezoelectric tab. In all studies on the high aspect ratio wing, it is demonstrated that structural flexibility can be exploited to increase aircraft performance. Other studies in this thesis investigate the applicability of robust control tools for flutter analysis considering model uncertainty and variation. First, different techniques for taking large structural variations into account are evaluated. Next, a high-fidelity numerical model of an aircraft with a variable amount of fuel is considered, and robust analysis is applied to find the worst-case fuel configuration. Finally, a study investigating the influence of uncertain external stores aerodynamics is presented. Overall, the robust approach is shown to be capable of treating large structural variations as well as modeling uncertainties to compute worst-case configurations and flutter boundaries.

Aeroelastic Concepts for Flexible Aircraft Structures 5

Dissertation

This thesis is based on a brief introduction to the area of research and the following appended papers:

Paper A

D. Eller and S. Heinze. An Approach to Induced Drag Reduction with Experi- mental Evaluation. AIAA Journal of Aircraft, 42(6):1478-1485, 2005.

Paper B

M. Jacobsen and S. Heinze. Drag Minimization of an Active Flexible Wing with Multiple Control Surfaces. TRITA-AVE 2006:50, Department of Aeronautical and Vehicle Engineering, KTH, 2006. Submitted for publication.

Paper C

S. Heinze and M. Karpel. Analysis and Wind Tunnel Testing of a Piezoelectric Tab for Aeroelastic Control Applications. AIAA Journal of Aircraft, 43(6):1799- 1804, 2006.

Paper D

S. Heinze and D. Borglund. Robust Flutter Analysis Considering Modeshape Variations. TRITA/AVE 2007:27, Department of Aeronautical and Vehicle Engi- neering, KTH, 2007. Submitted for publication.

Paper E

S. Heinze. Assessment of Critical Fuel Conﬁgurations Using Robust Flutter Analysis. Presented at the 47th AIAA/ASME/ASCE/AHS/ASC Structures, Struc- tural Dynamics, and Materials Conference, Newport, Rhode Island, May 2006. Submitted for publication.

Paper F

S. Heinze, U. Ringertz, and D. Borglund. On the Uncertainty Modeling for External Stores Aerodynamics. To be presented at the CEAS/AIAA/KTH Inter- national Forum on Aeroelasticity and Structural Dynamics, Stockholm, Sweden, June 2007.

Aeroelastic Concepts for Flexible Aircraft Structures 7

Contents

Preface 1

Abstract 3

Dissertation 5

Introduction 9 Aeroelasticity ...... 11 Aeroelastic phenomena ...... 11 Aeroservoelasticity ...... 14 Active Aeroelasticity ...... 15 Analysis and Testing ...... 16 Staticdeformations...... 17 Oscillations and ﬂutter ...... 19 Robust analysis ...... 21

Contribution of this work 23

Future work 25

References 27

Division of work between authors 31

Appended papers

Aeroelastic Concepts for Flexible Aircraft Structures 9

Introduction

Modern Formula One race cars, like the one shown in Figure 1, consist of highly optimized components with a lifespan hardly exceeding a single race. A lot of eﬀort is spent in research and development of improved components, often inspired by ﬁndings within aeronautical research. The aerodynamic

Figure 1: Formula One race car - in many ways related to aeronautics (photograph from BAE Systems). properties of such a car are taylored to control the airflow around the vehicle in order to obtain low aerodynamic drag and maximum downforce to provide the highest possible wheel traction. Aircraft wing airfoils, originally designed for producing lifting force, were found convenient for producing downforce by placing them upside down above the wheels. As for aircraft, however, increased normal force always comes at a cost of increased induced drag. Adjusting the wing shape and the angle of attack is thus a tradeoff between high downforce needed in turns and low drag desired on straight parts of the track, as shown in Figure 2. It is desirable to change the wing configuration during the race in order to get optimum performance in the entire range of operating conditions. In the 1990s, several racing teams began to use flexible wings, where the angle of attack decreased due to increased aerodynamic loading at increased velocities, 10 S. Heinze

Airflow Airflow Drag Drag

Downforce

High downforce Low drag Desirable in low−speed turns Desirable at high speed BUT: High drag BUT: Low downforce

Figure 2: Performance tradeoﬀ for wing design.

thus providing less downforce and less drag at higher speeds. Figure 3 shows the mechanism that is based on the exploitation of the aeroelastic interaction between the structure and the aerodynamic loads. The wing ﬂexibility is repre-

Zero airspeed Low airspeed High airspeed Undeformed Small deformation Large deformation

Figure 3: Flexible wing. sented by a spring element replacing the rigid attachment of the wing. The idea was to design a wing that would deform under aerodynamic loading, yielding an optimum wing shape at all speeds. At high speeds, the wing is deformed towards a lower angle of attack, thus reducing the drag in high-speed parts of the track, whereas the wing deforms less and maintains a higher angle of attack at low velocities. Unfortunately, the flexible wings caused some structural failures, and in the late 1990s, new regulations were issued that banned the use of elastic wing struc- Aeroelastic Concepts for Flexible Aircraft Structures 11 tures by restricting the maximum horizontal and vertical deformation of the wing under a given loading condition [1]. This simple example illustrates that flexibility can be exploited to gain performance of a structure with aerodynamic interaction, but also that flexibility introduces aeroelastic interaction that may cause the structure to fail.

Aeroelasticity

Aeroelasticity is the multidisciplinary science dealing with the interaction of aerodynamic forces and structural deformations. As a structure moves through the air, the motion will cause aerodynamic loads, leading to deformations of the structure. The deformation in turn has an impact on the airflow, thus changing the aerodynamic loading. Apparently, there is a closed loop of aerodynamic and structural interactions, and depending on the properties of the structure and the airflow, different phenomena can occur.

Aeroelastic phenomena Aeroelastic phenomena can be divided into different groups depending on the participation of specific members of the fluid-structure system. Figure 4 illustrates the disciplines that interact in aeroelasticity. The aeroelastic system

Aerodynamic properties

Static Flight aeroelsticity mechanics

Dynamic aeroelasticity Elastic Inertial properties Structural properties Vibrations

Figure 4: Aeroelasticity. 12 S. Heinze consists of three major components: aerodynamic forces due to the motion of the structure in the air, and elastic and inertial forces due to the structural deformation and acceleration.

Structural vibrations Combining elastic and inertial forces without aerodynamic interaction deﬁnes the equations of motion of the structure, where de- ﬂections and accelerations cause elastic and inertial forces, and an equilibrium of these forces leads to mechanical vibrations of the structure. Figure 5 shows an

Figure 5: Ground vibration testing of an ASK 21 glider aircraft at KTH. experimental setup where ground vibration testing is performed on an ASK 21 glider aircraft [2]. In this case, the aircraft was excited at speciﬁc frequencies and the response of the aircraft structure was measured using multiple accelerometers to determine the magnitude and the shape of the structural vibration. Ground vibration testing typically aims at identifying resonance frequencies and structural mode shapes in order to determine elastic and inertial properties of the structure prior to further investigations of the interaction with aerodynamic forces.

Flight mechanics Considering inertial and aerodynamic loads only, thus as- suming a rigid structure, defines the equations of motion in flight mechanics. Flight mechanics deals with the motion, stability and control of aircraft subject to different flight conditions in terms of altitude, airspeed, and aircraft weight. This subject deals with finding operational envelopes of aircraft, limited by properties such as the maximum lift provided by the wings or the maximum Aeroelastic Concepts for Flexible Aircraft Structures 13 thrust from the engines. Flight mechanical models are also used to design flight control systems, both for stabilizing the aircraft in flight and for optimizing the aircraft operation, for example in terms of minimizing fuel consumption for a given mission [3, 4, 5]. Although flight mechanics is not an aeroelastic discipline in itself due to the lack of elasticity, there may be a strong connection to aeroelasticity. For example, Figure 6 shows the NASA research aircraft Helios, for which the aerodynamic and inertial properties are strongly influenced by the elastic properties. Due to the significant wing bending, the rotational moment

Figure 6: The research aircraft Helios (NASA photo ED03-0152-4). of inertia and the center of gravity change significantly depending on the elastic deformation, which has to be accounted for in the design of the flight control system. This particular aircraft was actually lost due to excessive bending-pitch oscillations during a test flight [6].

Static aeroelasticity Combining aerodynamic and elastic forces according to Figure 4 defines the field of static aeroelasticity. This discipline deals with aeroelastic phenomena that deform the aircraft structure in flight in a quasistatic manner, thereby influencing aircraft performance and stability. Clearly, the performance of an aircraft depends on the aircraft shape. In cases where the shape of the aircraft has been designed for optimal performance without considering structural flexibility, it is likely that any deformation in flight will decrease performance, for example in terms of increased aerodynamic drag or decreased control surface efficiency. It is therefore essential to estimate the aerodynamic forces and the resulting structural deformations in early stages of the aircraft design process. Clearly, aerodynamic loads change during flight, 14 S. Heinze and thus a structure has to be found that performs reasonably well within the entire range of flight conditions. Besides the impact of static deformations on the performance, there may even occur instabilities due to the interaction between the aircraft flexibility and the quasistatic airloads. A phenomenon called divergence occurs when the loads deform the structure in a way that increases the aerodynamic loading, thus deforming the aircraft even more, finally leading to structural failure. Al- though this phenomenon is not truly static, it is usually classified as a static phenomenon due to a relatively slow process without oscillations and without any significant impact of the inertial forces.

Dynamic aeroelasticity Dynamic aeroelasticity combines all the disciplines involved in aeroelasticity. The most well-known dynamic aeroelastic phenomenon is flutter, an interaction between structural vibrations and unsteady aerodynamic forces, where oscillations increase in magnitude, eventually leading to structural failure. The aerodynamic forces usually introduce damping to the structural oscillations once the structure starts moving in the airflow, but for increasing velocity, the phase difference between the inertial, elastic, and unsteady aerodynamic forces may cause the total damping to decrease, and finally vanish. This point defines the flutter speed, and aircraft designers need to ensure that the flutter speed lies outside the flight envelope of the aircraft [7]. Apart from the unstable flutter phenomenon, dynamic aeroelasticity also includes phenomena such as turbulence induced vibrations in flight, gust loads, and limit-cycle oscillations, being a type of flutter where some part of the aeroelastic system restricts the magnitude, thus making the flutter oscillations stable.

Aeroservoelasticity The subject of aeroservoelasticity deals with the interaction between aeroelastic phenomena and the flight control system (FCS). Highly flexible aircraft in particular are subject to large deformations that must be taken into account in the design of the FCS, as mentioned earlier for the Helios aircraft. The design of the FCS has to ensure that the controller does not introduce instability in the aeroservoelastic system. On the other hand, the FCS can also be used to stabilize an aeroelastic system that by itself would be unstable. Provided that the control system and the actuators are sufficiently fast, the FCS may be used to stabilize flutter oscillations, thus making it possible to operate an aircraft beyond the critical flutter speed [8]. Less critical aeroservoelastic applications are the suppression of vibrations and gust loads, as in Paper C of this thesis, where the flexible high aspect ratio wing shown in Figure 7 is considered. In this study, it is shown that a Aeroelastic Concepts for Flexible Aircraft Structures 15 piezoelectric tab can be used to alleviate gust loads significantly by designing a control system that accounts for the aeroelastic properties of the wing.

Figure 7: The high aspect ratio ﬂexible wing mounted in the low-speed wind tunnel L2000 at KTH.

Active Aeroelasticity Due to the stability and performance issues related to aeroelasticity, this subject has long been seen as a problem in aeronautics. Before, aircraft designers had to make sure that the aircraft was free from aeroelastic instabilities and provided enough stiffness to keep deformations within reasonable limits [7]. As new tools for aeroelastic analysis of aircraft structures have evolved, the aeroelastic behavior has successively become part of the aircraft design process [9, 10, 11]. Nowadays, the aeroelastic behavior of a flexible structure can be optimized, and the flexibility can be taken advantage of. The objective of active aeroelasticity is to exploit the aeroelastic interaction by controlling the airloads in order to obtain some desired structural deformation. Aircraft performance may thus be increased when allowing for more flexibility of the structure, since deformations then can be controlled more easily. 16 S. Heinze

One well-known research project dedicated to active aeroelasticity is the Active Aeroelastic Wing (AAW) program in the USA [12]. The AAW program considers a modified F-18 fighter aircraft, where the torsional flexibility of the wing is exploited by using leading-edge control surfaces to twist the wings in flight. In this particular case, the flexibility of the wing first turned out to be a problem, since the efficiency of the trailing edge control surfaces decreased significantly at higher airspeeds due to the elastic deformation of the wing. Using the leading-edge control surfaces, however, it was demonstrated that the wing flexibility could increase the efficiency of the controls and thus increase the maneuverability of the aircraft. A similar study has also been performed at KTH, where the flexible wing shown in Figure 8 was studied [13]. In this study, it was demonstrated how the

Figure 8: Flexible wing mounted in a low-speed wind tunnel at KTH. torsional flexibility could be exploited in the development of control laws for efficient multi-surface roll control of the model. Another study aiming at exploiting aeroelastic effects is presented in Paper A, where the wind tunnel model in Figure 7 was controlled using 10 leading edge and 10 trailing edge flaps. The aim of the study was to deflect the control surfaces such that the resulting aeroelastic deformation gave an optimal lift distribution that minimized the induced drag. This study was part of the European research project Active Aeroelastic Aircraft Structures [14] focusing on novel concepts for flexible aircraft. Within this project, different configurations have been considered and active aeroelastic concepts evaluated [15, 16, 17].

Analysis and Testing

Due to the multidisciplinary nature of aeroelasticity, the analysis tools com- bine many different subjects such as aerodynamics, structural dynamics, rigid Aeroelastic Concepts for Flexible Aircraft Structures 17 body mechanics and control theory [18, 19]. Depending on the phenomenon or mechanism in focus, different combinations of these disciplines are used in the analysis. In the same manner, experimental setups for investigating phenomena in aeroelasticity can be very different depending on the subject in focus. Ex- periments focusing on the stiffness properties, for example, may be based on some structure subject to static external loading, whereas flutter investigations involve flight testing or wind tunnel experiments [20]. In this thesis, analysis and testing focusing on the static deformation under aerodynamic loading as well as analysis and testing of aeroelastic oscillations and flutter are presented. Analysis and testing are two different approaches to investigate aeroelastic phenomena and usually complement each other. Depending on the phenomenon that is investigated, however, experiments may be expensive or even impossible to perform, for example when investigating stability limits of aircraft. In this case, it is essential to have reliable analysis tools to investigate these phenomena. On the other hand, the numerical model used in the analysis is often based on results from experimental investigations, and in cases where the numerical model contains uncertainties, experimental data can be used to improve the quality of the model. In particular when novel theoretical approaches are evaluated, it is essential to validate numerical results by comparing them to test data. In this thesis, most of the studies involve both analysis and testing. The combination of both numerical and experimental results is important since it may both prove the reliability of the numerical models and tools and enable the detection of errors and uncertainties in the analysis.

Static deformations During flight, aircraft most of the time operate under steady-state conditions, where the aerodynamic loads cause static deformations of the aircraft structures. Theoretical tools for estimation of the static aeroelastic deformation are required, and wind tunnel experiments can be used to validate the numerical predictions. The theoretical approach for computing aeroelastic deformations is to develop a mathematical model describing the structural behavior such that deformations can be computed from external forces. This model is coupled to another model for the aerodynamic properties, where aerodynamic loads are computed depending on the flight condition and the structural shape. In wind tunnel testing, deformations can, for example, be measured optically [21, 22]. In paper A, an approach is presented where the deformation of the wing in Figure 7 is controlled to obtain minimum induced drag for a range of flight conditions. The wing deformation is estimated numerically by coupling a finite element model of the wing structure [23, 24] to aerodynamic forces 18 S. Heinze obtained from a boundary element method [25]. In the experiments, the deformation is measured optically using four CCD cameras as shown in Figure 9. The

Figure 9: Optical deformation measurement using CCD cameras.

QualiSys camera system [26] sends infrared flashes to the wind tunnel model that are reflected back to the cameras from a set of reflecting markers. The cameras monitor two-dimensional positions of the markers, and compute three- dimensional locations by combining the two-dimensional pictures. In Paper A, the wing deformation could thus be measured and compared to numerical predictions, both to validate the numerical model and analysis tools, and to validate the concept of controlling the lift distribution of the flexible wing for minimum induced drag. In Paper B, the static deformation was exploited to obtain minimum total drag of the same wing configuration. In this case, however, the deformation was neither estimated numerically nor measured in the experiment. Instead, the drag Aeroelastic Concepts for Flexible Aircraft Structures 19 was measured using a wind tunnel balance in the wing root [20], and a real-time optimization algorithm was applied to minimize the measured drag during the experiment. Clearly, the drag is a function of both the wing deformation and the control surface deflection, and it was demonstrated that the drag can be minimized without knowledge of the elastic properties and deformation of the wing.

Oscillations and flutter Structural oscillations in flight typically decrease the ride quality and wear out the structure. In particular when the aircraft approaches the flutter limit, oscillations excited by turbulence or gusts may have significant magnitudes and lead to high structural loading. Most importantly, it is vital to design aircraft such that the flutter speed lies well above the operational envelope of the aircraft. Also, in order to avoid excessive structural loading, the aeroelastic damping of the aircraft should be sufficiently high to make structural oscillations decay quickly in the entire operational range of the aircraft. In aircraft design, analysis has to be performed to ensure sufficient aeroelastic damping. Flutter has been in the focus of aeronautical research since the 1930s [27], when aircraft aiming at higher speed and lower weight suffered from flutter instabilities and were damaged or destroyed. Since then, theories for computing unsteady airloads have evolved to increase the accuracy of the predictions, and to allow for modeling of more complex geometries as well as analysis in the transonic and supersonic velocity range [28, 29, 30, 31, 32]. Due to the desire of modeling complex geometries accurately, the computational effort for these analyses naturally increases. Significant research effort has been devoted to the development of efficient analysis tools that reduce the computational effort, yet maintaining high accuracy.

Time domain approach One approach for numerical investigations of dynamic aeroelastic phenomena is time-domain analysis, where the structural motion and the airloads are computed by stepping in time [10, 25, 33, 34]. The accuracy of the time-domain approach can typically be improved by increasing the fidelity of the numerical model, however at higher computational cost. Even nonlinearities such as freeplay and friction can be modeled easily and accounted for in the analysis. Today, significant research effort is spent on making time-domain computational aeroelasticity more efficient [35, 36, 37].

Frequency domain approach Another approach for simplifying the analysis is to use frequency-domain analysis, where the motion of the structure is assumed 20 S. Heinze to be a combination of harmonic eigenmodes. This approach is based on linear structural dynamics and aerodynamics, which is usually sufficiently accurate for small deformations. When nonlinearities such as freeplay or friction are present, however, the linear approach is not useful. Unfortunately, freeplay of for example control surfaces may have significant impact on the flutter behavior, and thus the linear frequency domain approach may be insufficient in some cases. In this thesis, the flutter analyses presented in Papers D, E and F are based on the linear frequency-domain approach. In Paper F, both analysis and experiments are presented for the delta wing configuration shown in Figure 10. In this

Figure 10: Delta wing mounted in wind tunnel. study, the numerical capabilities of the frequency-domain approach are investigated by considering an aircraft geometry with different levels of complexity. Due to the risk for structural damage and failure, the flutter limit of aircraft is usually not investigated in flight tests. The wing shown in the figure was however designed to operate under flutter conditions with significant structural oscillations. The wing was tested at different airspeeds up to the flutter speed, and results in terms of damping and frequency of the dominating vibration were compared to numerical predictions. Aeroelastic Concepts for Flexible Aircraft Structures 21

Robust analysis Due to the simplifications in modeling and analysis of aircraft structures, there is always some uncertainty in the numerical models leading to deviations between analysis and test results. One way to enhance the results is clearly to increase the fidelity of the numerical model or to improve the method used for solving the problem. Another possibility is to consider the uncertainty in the model, and to incorporate this uncertainty in the numerical model. In particular regarding the flutter limit, where experiments cannot be performed due to the high risk involved, the question arises how uncertainty in the model will influence the predicted flutter speed. It would be desirable to compute flutter limits that are robust with respect to the uncertainty in the nominal analysis. Robust approaches have been presented in the past, where tools from the control community [38] are applied to find robust flutter limits based on mathematical descriptions of the uncertainty present in the numerical model, see Refs. [39] to [48]. It is essential that the uncertainty description captures the real uncertainty present in the system, which clearly is difficult since the uncertainty is not known. Approaches have been developed where experiments at subcritical conditions are considered to estimate the uncertainty, and robust analysis is performed to compute the effect of this uncertainty on the flutter speed [49, 50]. In Paper F, an approach is used where new validation techniques are used to estimate the magnitude of the uncertainty, given that the basic uncertainty mechanism is known. The robust approach can also be applied when considering variations instead of uncertainties in the model. Uncertainties and variations are described and treated in a similar way in robust analysis, the main difference being that the variation parameters are explicitly known, for example when describing the fuel level in an aircraft tank. Papers D and E demonstrate how the robust approach can be applied to find worst-case conditions from a range of possible aircraft configurations.

Aeroelastic Concepts for Flexible Aircraft Structures 23

Contribution of this work

This thesis includes studies from different disciplines within the general area of aeroelasticity. Three studies focus on active aeroelastic concepts for improved aircraft performance by exploitation of structural flexibility, and three studies focus on robust flutter analysis considering structural variations and aerodynamic uncertainties. Papers A and B demonstrate that the aerodynamic drag, and thus the fuel consumption, can be reduced by using redundant control surfaces. Even though the considered wind tunnel model has unrealistically many control surfaces, it was found that using only a few of them could improve drag performance significantly. In principle, modern aircraft could use the drag minimization algorithm presented in Paper B without the need of additional hardware in the aircraft, since redundant control surfaces are usually present. In particular when a fly-by-wire flight control system is used, it may be possible to implement the drag minimization algorithm in the flight control system. Paper C demonstrates that active control can be applied to improve the gust response and thus the ride quality of a flexible wing by using a relatively small piezoelectric tab. The focus of this study is on the concept of aeroelastic amplification, where the structural flexibility and the airloads are exploited to increase the efficiency of a rather small control surface. Papers D and E demonstrate the application of robust flutter analysis to aircraft configurations with varying structural properties. Paper D focuses on the method and investigates the accuracy of the robust analysis when large variations are present. Also, this study presents different means to account for large structural variations in the model. Paper E focuses on the practical application of the developed approach by considering fuel variation in a realistic high-fidelity aircraft model. The robust approach is found to be capable of detecting worst-case fuel configurations with respect to the flutter speed. Paper F demonstrates how the deviation between experimental and predicted data may be used to estimate uncertainty descriptions that can be used in robust analysis. The focus is on successively increasing the geometrical complexity of the experimental setup, and evaluating how accurately the numerical model can predict the behavior of the different configurations. This paper also demonstrates that small details can have significant impact on the flutter behavior of a wing, and it is found that accurate uncertainty models capturing the true uncertainty mechanisms are essential.

Aeroelastic Concepts for Flexible Aircraft Structures 25

Future work

The studies on drag minimization indicated that drag reductions may be possible for operational transport aircraft. In the 1990s, NASA applied a similar approach in flight testing of a transport aircraft [51], where one variable was used to minimize the aerodynamic drag in flight. It would be interesting to apply the presented drag reduction approach to a real aircraft model, identify redundant control surfaces and thus optimization variables, and demonstrate the applicability of the multivariable real-time drag-reduction algorithm. Regarding the piezo tab study, all experiments performed aimed at validat- ing the numerical model used in the analysis and design of the load-reducing control system. Unfortunately, wind tunnel experiments evaluating the control system capabilities could not be performed due to the lack of experimental equipment, such as a gust generator. The piezo tab study could thus be continued by implementation of a gust generator device to validate the control system performance in experiments. Research on robust analysis and optimization is still ongoing. The presented papers demonstrate the usefullness of the robust analysis tools, but they also demonstrate the need for accurate uncertainty models to guarantee robust, but not excessively conservative results. The identification of typical uncertainty mechanisms and the development of appropriate uncertainty descriptions are thus of vital importance to future developments.

Aeroelastic Concepts for Flexible Aircraft Structures 27

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Division of work between authors

Paper A

Heinze designed, manufactured and instrumented the experimental setup and developed the analysis model of the structure, as well as the mixed experimental/numerical technique for determination of the induced drag in the experiment. Eller performed the aerodynamic modeling and the optimization study. The experiments were performed by Heinze. The paper was written by Eller with support from Heinze. Paper B

Heinze designed, manufactured, and instrumented the wind tunnel model. The drag minimization algorithm was developed and implemented by Jacobsen. Ex- periments were performed jointly by the authors. The paper was written by Jacobsen with support from Heinze. Paper C

Heinze designed, manufactured and instrumented the experimental setup and derived the numerical analysis model. The analysis model was validated experimentally by Heinze. Karpel modiﬁed the analysis model and performed the control law design. The paper was written jointly by the authors. Paper D

Borglund provided the baseline model used in the test case and a basic version of the robust ﬂutter solver. Heinze extended the numerical model to include uncertainty in the mass distribution and performed all analysis presented in the paper. The paper was written by Heinze with support from Borglund. Paper F

Heinze designed and manufactured the experimental setup. The nominal numerical models were developed by Heinze and Ringertz. Borglund provided tools for robust analysis. Heinze performed nominal and robust analysis and performed the experiments. The paper was written by Heinze with support from Borglund and Ringertz.

Paper A

An Approach to Induced Drag Reduction with Experimental Evaluation

David Eller and Sebastian Heinze Royal Institute of Technology SE-100 44 Stockholm, Sweden

Abstract An approach to minimize the induced drag of an aeroelastic configuration by means of multiple leading and trailing edge control surfaces is investigated. A computational model based on a boundary-element method is constructed and drag-reducing flap settings are found by means of numerical optimization. Further, experiments with an elastic wind tunnel model are performed in order to evaluate the numerically obtained results. Induced drag results are obtained by analysing lift distributions computed from optically measured local angles of attack, since standard techniques proved insufficient. Results show that significant reductions of induced drag of flexible wings can be achieved by using optimal control surface settings.

Introduction

Optimization of aerodynamic drag is commonly concerned with finding the optimal shape of a body with minimal drag for a specified, fixed flight condition. Since the aircraft shape is not usually assumed to be variable in operation, some sort of compromise has to be found for vehicles which need to operate efficiently within a wide range of lift coefficients, Mach numbers or altitudes. Examples of such aircraft may be • commercial transports with extremely long range, where the aircraft weight changes considerably during cruise due to fuel consumption,

• long-range unmanned aerial vehicels with a possibly even higher fuel weight fraction, or

• vehicles with wide operating envelopes which may need to operate and maneuver at widely varying speeds and altitudes.

A 1 A 2 D. Eller and S. Heinze

In such cases, a system which could minimize induced drag within most or all of the operational envelope would be beneficial. Clearly, optimization of the external shape alone cannot provide optimal performance for a wide range of operational parameters, especially when considering aeroelastic deformations [1]. Another aspect of induced drag minimization is that maneuver performance can be considerably improved for some vehicles. Light-weight long-range aircraft tend to have rather limited excess power to overcome the additional induced drag incurred by the increased lift necessary for maneuvering [2]. Means to adaptively provide a wing configuration with minimum induced drag for maneuvering conditions could therefore improve operational flexibility. In order to obtain efficient baseline performance, vehicles of the type mentioned above will likely feature light-weight high aspect ratio wings leading to considerable flexibility and corresponding aeroelastic deformations even in cruise flight. Since the influence of this deformation on the spanwise lift distribution can be very considerable, it needs to be included in an analysis of induced drag. In this context, ‘induced drag’ refers to the drag caused by the spanwise distribution of lift, and does not include other drag components which may also depend on lift to some degree. The most straightforward method to enable drag optimization under different flight conditions is the provision of a certain number of conventional control surfaces at the leading and trailing edges of the lifting surfaces [1]. Vari- able camber wings or controllable wingtip extensions are interesting alternative concepts which may require more system integration efforts [3, 4]. Flap-based active aeroelastic control has been investigated [5, 6] and successfully evaluated experimentally for maneuverability improvements and load alleviation, although usually with a fixed set of control surfaces [3, 7]. Kuzmina et al. [8] presented a numerical study showing the possibility of reducing induced drag of elastic aircraft using control surfaces. The aim of this study is to investigate which and how many control surfaces are required to achieve a worthwhile reduction in induced drag for a highly flexible lifting surface. In particular, the optimal configurations determined numerically are to be tested in the wind tunnel in order to evaluate if the computed drag reductions can be obtained. Experimental determination of induced drag requires a method to separate the induced drag component from the measured total drag. Since induced drag constitutes only a certain fraction of the overall aircraft drag, employing an optimal setup of deflected control surfaces may or may not reduce total drag. Possible sources of additional viscous drag could be premature transition triggered by upstream control surfaces or flow separation caused by the disturbance of the chordwise pressure distribution. However, even airfoils Induced Drag Reduction with Experimental Evaluation A 3 for low Reynolds number flow can be designed for extremely low profile drag at trailing edge flap deflections of up to 15◦ (see e.g. Althaus [9]). It should hence be possible to achieve reasonable values of viscous drag coefficients at least within a range of moderate control surface deflections.

Test case

Within the framework of the European Union project Active Aeroelastic Aircraft Structures (3AS, Schweiger and Suleman [10]), a wind tunnel model has been designed and built for aeroservoelastic investigations. The 1.6 m semispan model consists of a load-carrying beam of carbon-fiber composite and ten aerodynamic sections rigidly clamped to the beam in a manner which prevents them from contributing stiffness to the beam structure. The beam can easily be replaced to allow for modification of the stiffness properties. Each segment is fitted with leading- and trailing edge flaps, controlled by two miniature electric actuation devices driving an internal geared transmission for high deflection angle accuracy. The airfoil used in a related full-scale design study, with 16% thickness and 3.6% camber, is also used for the model sections. Designed for velocities above Mach 0.5 and Reynolds numbers in the range of 107, this wing is not expected to perform optimally in terms of profile drag under low-speed wind tunnel testing conditions with Reynolds numbers of about 3 · 105. In order to avoid excessive profile drag, flap deflections are therefore limited to moderate angles of ±10◦ . With a corresponding full-span aspect ratio Λ of 20 and 10◦ wing sweep, the wing model is considered to be a reasonable approximation of the generic high-altitude long-endurance UAV under investigation by 3AS project partners. A section of the wind tunnel model showing the arrangement of the structural beam and the control surfaces is shown in Figure 1. Leading edge flaps have a depth of 20% chord, those at the trailing edge 25%. The spanwise slots between the rigid shell and the control surfaces are sealed using elastic polymer strips which are used for the same purpose on high-performance gliders. The elastic axis is located at 36% chord, in the center of the load carrying beam. The structural properties of the beam vary along the span of the wing. There are four regions where the beam thickness and thus stiffness is kept constant. The first region with the highest stiffness covers four inboard sections. The remaining regions are covering two sections each, with decreasing beam thickness as the span coordinate increases. The laminate is free of membrane-bending coupling. Stiffness values are shown in Table 1. A 4 D. Eller and S. Heinze

Figure 1: Sectional view of the wind tunnel model sections.

Section 1-4 5-6 7-8 9-10 EI [Nm2] 437.8 299.6 156.7 31.1 GJ [Nm2] 35.7 23.5 11.7 5.7

Table 1: Stiﬀness distribution along the span.

Numerical modeling

Instead of a full-scale wing, the wind-tunnel model itself was modeled for the numerical optimization process in order to simplify comparison with experimental data. In the following, the modeling approach used to obtain the objective function and constraints for the optimization problem is described.

Structure The structural model used for numerical simulations was validated experimentally. The preliminary stiffness distribution that was derived from composite material properties and static experiments was updated by matching experiments with static loading and measured eigenfrequencies. A comparison of predicted and measured eigenfrequencies is shown in Table 2. Especially in the lower frequency range, the measured data fits very well with the predictions. Mass properties were not updated when matching the model to the vibration testing data, since the mass distribution could be determined analytically with high accuracy. Experiments showed furthermore that the flexibility of the wind tunnel mounting about the x-axis had to be included in order to correctly model the structural behavior. From experimental measurements, the relevant rotational Induced Drag Reduction with Experimental Evaluation A 5

Mode fexp [Hz] fpred [Hz] Error [%] 1st bend 3.2 3.14 −1.9 1st in-plane 6.0 6.06 1.0 2nd bend 14.8 14.5 −2.0 1st tor 17.4 17.0 −2.3 2nd in-plane 32.9 32.3 −1.8 3rd bend 33.6 32.8 −2.4 2nd tor 39.0 39.8 2.0 4th bend 54.1 56.8 4.9 3nd in-plane 82.6 78.3 −5.3

Table 2: Vibration testing results. stiﬀness around the x-axis was found as 1950 Nm/rad. In the ﬁnite element model, the mounting was included as a torsional spring between wing root and wind tunnel wall.

Aerodynamics An unstructured boundary element method is used to compute the aerodynamic loads for the elastic wind tunnel model. While the method employs an approach generally similar to conventional three-dimensional panel methods, it allows for simulations with moving and deforming meshes and features a variant of the Kutta condition suitable for conﬁgurations with rotating control surfaces. Aero- dynamic forces and moments are obtained from surface pressure integration. Some details on the boundary element method are given below, more information can be found in Eller and Carlsson [11].

Flow model. The method solves the linearized equations of potential flow. The wetted surface of the body is discretized and a piecewise linear distribution of source and doublet potentials is assumed on the surface. The boundary condition of flow parallel to the surface can be written as a Dirichlet condition for the potential at each collocation point, which leads to a system of linear equations. While the general approach differs little from the conventional panel methods described in Katz and Plotkin [12], the numerical solution makes use of panel clustering [13] in order to improve the algorithmic scaling properties. By employing a preconditioned iterative linear solver, very large problems can be solved with moderate computational cost. The method differs from the Doublet Lattice Method (DLM) [14] often used in aeroelasticity in that the true surface A 6 D. Eller and S. Heinze

Figure 2: Shape of the complete deformed wing. Insert shows coarse mesh of the wing tip region with the ninth trailing edge flap deflected 10◦ down. shape including wing thickness is geometrically modelled. It is hoped that the higher geometric fidelity improves the prediction of control surface effects. Boundary layer displacement, however, is not accounted for yet.

Discretization. Unstructured triangular surface meshes are used in order to allow the treatment of complex geometries. Since the geometry of the wind tunnel model considered here is rather simple, triangulated structured meshes are used to reduce the number of surface elements. Control surface motion is modeled by rotating appropriate subsets of the surface mesh around the corresponding hinge axis. The gaps which would normally open to the left and right of the moving surface elements are not modeled; instead, some surface elements are stretched to cover the gaps. This approach yields satisfactory meshes for flap deflection up to at least 15◦ without special mesh refinement, which is more than sufficient for the current study. In Figure 2, the overall geometry of the surface used in the simulation is shown, together with a detailed view of the mesh in the vicinity of a deflected control surface. Symmetric flow is enforced by modeling the half-wings on both sides of the symmetry plane.

Kutta condition. As the solution of the governing equation of steady potential flow alone cannot yield a resulting force on a closed body, a wake surface carrying doublet singularities must be modeled. The strength of these doublets is determined by introducing additional equations which enforce the Kutta condi- Induced Drag Reduction with Experimental Evaluation A 7 tion for the resulting flowfield. However, since the Kutta condition is empirical, different mathematical formulations exist. When the condition is enforced in terms of a vanishing vortex strength along the trailing edge [12], or as a function of flow velocity components normal to the surface, large velocity gradients can result when the trailing edge is not smooth along the span. Alternatively, the Kutta condition can be expressed in terms of tangential velocity components. As the surface tangent vectors in the downstream direction do not change as rad- ically as the surface normals in the flap region, smooth pressure distributions are obtained.

Force integration. Some difficulty is introduced by the relatively poor ‘condi- tioning’ of the drag computation by pressure integration. Since the drag force is a fairly small component of the resulting aerodynamic forces acting on the wing, and thus a small difference of fairly large pressure forces acting in different directions, small relative errors in surface pressures can lead to significantly larger relative drag errors. For the coarsest mesh used in this study, with about 4000 surface elements, the absolute error in induced drag coefficient was found to be approximately 0.003 when compared with measurements. In general, this is a fairly good accuracy when compared to drag coefficients for complete configurations. Unfortunately, for a very high aspect ratio wing, this error constitutes a sizeable fraction of the induced drag at low lift coefficients.

Boundary layer effects. As mentioned before, wind tunnel testing was restricted to rather low Reynolds numbers in the order of 3 · 105. For such conditions, viscous effects play an important role. For example, laminar separation bubbles are relatively likely to occur [15]. While this study is not concerned with the associated increase in profile drag, even small separations can lead to a pressure distribution which differs somewhat from the inviscid one computed by the numerical method. Furthermore, turbulent boundary layers grow rather quickly at low Reynolds numbers and can reach a thickness of several percent chord at the trailing edge. Lacking symmetry between upper and lower airfoil side, such thick boundary layers usually reduce the effective camber of the wing, changing both lift and section pitch moment. Naturally, these issues affect the comparison of experimental and computed results negatively. In earlier work [11], with Reynolds numbers exceeding 106, considerably better agreement could be obtained. Nevertheless, the current method achieves sufficient accuracy in predicting the spanwise distribution of loads and deformations for a given total lift, so that the induced drag optimization can be performed. At Reynolds numbers more typical for full-scale applications, boundary layer thicknesses tend to grow more slowly and separa- A 8 D. Eller and S. Heinze tion bubbles are less likely, so that better agreement would be expected.

Aeroelastic coupling For the structural model, a simple beam approximation was chosen since the actual load-carrying structure is long and slender. A Nastran [16] ﬁnite-element model was constructed for the internal wing beam. From this, a reduced stiﬀ- ness matrix K could be obtained for the set of ten locations where the rigid aerodynamic sections are attached. Pressure values from the aerodynamic analysis are integrated to obtain forces and moments for each of the wind tunnel model sections. These loads are applied as point loads acting on the section at- tachments, from which beam deformations are computed. Beam deformations u can then be mapped to surface vertex displacements using an interpolation procedure. To solve the static aeroelastic problem

Ku = Fa(u, δ), (1) where Fa contains the aerodynamic loads on the model segments, depending on the beam displacements u and the vector of control surface deﬂections δ, a damped ﬁxed-point iteration is employed:

k+1 k −1 k u = (1 − ω)u + ωK Fa(u , δ). (2) With a damping factor ω = 0.7, the simple iteration method converges within four to six steps, which makes it computationally eﬃcient. As the condition for convergence, the relative change of the surface doublet strengths between consecutive iterations was required to fall below 1%,

|µk − µk−1| < 0.01 |µk|, (3) where µk is the vector of doublet strengths at step k. A similar criterion can be defined for the deformations u, but (3) was found to be somewhat more strict since doublet strengths converged at a slower rate than deformations. Problem (1) can also be treated by computing the Jacobian ∂F J = K − a , (4) ∂u and applying Newton’s method. Since, for small deformations u, aerodynamic loads are almost linearly dependent on u, a single step is often sufficient. For linear aeroelastic stability problems, where infinitesimally small deformations are considered, this approach is accurate and efficient. With the current boundary element method however, the computation of J is much more costly than Induced Drag Reduction with Experimental Evaluation A 9 a few iterations of (2), which only take a few seconds each. For high airspeeds close to static aeroelastic divergence, J becomes nearly singular and the rate of convergence for a simple method as (2) would degrade rapidly. In that case, a direct solution (or Newton’s method for large deformations) is usually more efficient.

Mesh convergence. In order to evaluate if the simplified aerodynamic model is sufficiently accurate, numerical results for static equilibrium deformations are compared with measured data in Figure 3. It should be noted that the optical measurement system tracks marker positions only, so that twist values given in Figure 3 are computed from differences of marker displacements and hence are less accurate than translational deflections. Tip deflections are relatively large, ◦ reaching 7.5% of the semispan and about 4 at the wing tip for CL = 0.5 at 30 m/s. The coarse mesh with 4020 surface elements matches the measured twist deformation well, and overpredicts bending deflections only marginally. The finer meshes with 7060 and 11220 elements, respectively, are slightly closer to the measured bending deflections, but overpredict twist instead. From earlier studies, it is known that normal forces converge rather quickly with mesh refinement, while section pitch moments require slightly higher mesh resolution for good accuracy. Therefore, it must be concluded that the finer meshes represent more accurate solutions of the linear potential flow model, but that the actual twist moments encountered in the wind tunnel are smaller than the flow model predicts. The optimization described later is performed with the coarse mesh, since, in this particular case, it actually predicts twist deformations better than a numerically more accurate solution of the potential flow equations.

Optimization

Two diﬀerent formulations of the drag minimization problem were tested in this study. While both approaches were tested, results are presented only for the second alternative which proved superior for the problem considered. In the ﬁrst approach, the drag minimization was formulated as a straight-forward nonlinear programming problem of the form

minimize CDi(δ) (5)

subject to CL ≥ CL,ref (6)

and δl ≤ δ ≤ δu, (7) A 10 D. Eller and S. Heinze

0.12 experiment 0.1 coarse mesh intermediate 0.08 fine mesh 0.06

0.04 Deflection [m] 0.02

0 0 0.25 0.5 0.75 1 1.25 1.5

−1

−2

−3 Twist [deg] −4

−5 0 0.25 0.5 0.75 1 1.25 1.5 Span coordinate [m]

Figure 3: Comparison of static aeroelastic deformation.

where CDi is the coefficient of induced drag, CL the lift coefficient constrained to be larger than some reference value CL,ref and δ the vector of nd design variables holding the deflection angles of the control surfaces, bounded by the lower and upper limit angles δl and δu. The drag coefficient CDi is in this case computed by pressure integration. Since the lift coefficient depends approximately linearly on flap deflections for the range of deflection angles considered, the lift constraint (6) can be expressed as a linear inequality

∇δCL · δ ≥ CL,ref − CL|δ=0, (8) with the vector of control surface derivatives ∇δCL. Formulating the lift constraint in a linear manner leads to reduced computational cost in the optimization process. The nonlinear programming problem (5)-(7) was solved approximately using the quasi-Newton method built into Matlab [17], which evaluated the objective Induced Drag Reduction with Experimental Evaluation A 11 function between 40 and 500 times, depending on the number of design variables. Solutions obtained in this way yield flap settings which reduced the computed induced drag. The weak point of this approach was found to be the drag computation by pressure integration, which required a very fine mesh for accurate results. With moderate resolution like those used to test this optimization approach, the gradients ∇δCDi computed by the optimization software in each iteration were not sufficiently accurate, so that the quasi-Newton iteration converged toward points which were not really optimal. With better mesh resolution, computational costs became unacceptable due to the large number of function evaluations. In contrast to induced drag, normal forces can be computed with reasonable accuracy even with rather coarse meshes. In order to allow for a more efficient formulation of the optimization problem, the computed spanwise distribution of circulation Γ is compared with an elliptic distribution Γe, which is known to be optimal for a subsonic wing-only configuration. Instead of a nonlinear programming problem, a least squares problem of the form

2 minimize |Γ(δ, yi) − Γe(yi)| (9)

subject to δl ≤ δ ≤ δu (10) is obtained. Here, the lift distribution is expressed in terms of the distribution of circulation strengths Γ(δ, yi) at a number of span coordinates yi. Due to the approximate linearity of the lift distribution with respect to flap deflections, the least squares problem can efficiently be solved in its linear form, where

Γ(δ, y) ≈ Γ0(y) + ∇δΓδ. (11)

In this case, only the circulation distribution Γ0(y) for a reference case and the Jacobian ∇δΓ with respect to flap deflections need to be computed. The reference case would usually be the configuration with neutral flap settings, where the required lift coefficient is achieved by setting an incidence angle at the wing root. Lift constraints are not necessary in this formulation since an optimal solution approximates the elliptic distribution with the correct total lift as closely as possible. This approach resembles the drag minimization procedure of Kuzmina et al.[8] in that it uses the circulation distribution. The difference is that, here, the induced drag is not computed from Γ within the optimization. Instead, the circulation distribution is used to form a more readily solved least squares problem, which is possible in the present case. When formulating the problem according to (9)-(10), a relatively coarse mesh will suffice as only the lift distribution is needed, not the integrated drag value. Furthermore, computing the Jacobian by finite differences is possible using only A 12 D. Eller and S. Heinze

nd + 1 solutions of the static aeroelastic problem, as opposed to the nd + 1 solutions required to compute the gradient in each iteration of a nonlinear optimization solver, where nd is the number of design variables. The number of design variables can be varied while the physical model with its 20 control surfaces per semi-span is left unchanged, simply by deflecting groups of control surfaces together. With 10 design variables, pairs of flaps (two leading-edge or two trailing-edge surfaces) are deflected to identical positions, while with four variables, outboard groups of 4 and inboard groups of 6 surfaces are moved to the same deflection. Figure 4 shows a sketch of the configurations investigated. Note that, in the case with n = 3, only trailing edge surfaces are deflected, but the pattern of flap deflections is identical to the case with six variables. The angle of attack at the wing root α is an additional design

Figure 4: Control surface configurations. variable since a change in α induces different changes in spanwise lift and moment distribution than flap deflections. However, it is likely that the parasitic drag of the fuselage and other non-lifting aircraft components would increase strongly, should the angle of attack deviate substantially from some optimal range. Therefore, α is constrained to lie between the somewhat arbitrary bounds −2◦ and 4◦ . In some cases, optimization resulted in flap settings featuring large differences in deflection for neighboring control surfaces. It is likely that such a configuration would increase viscous drag substantially. Therefore, additional constraints were introduced in (9)-(10) in order to avoid large differences in deflections. For pairs of neighboring flaps (i, i + 1), two conditions of the form

◦ δi − δi+1 ≤ 4 (12) ◦ δi+1 − δi ≤ 4 (13) are imposed. The limit 4◦ was chosen rather arbitrarily and was intended to be Induced Drag Reduction with Experimental Evaluation A 13 conservative in order to evaluate the impact of this type of constraint on the optimal result. However, the optimal induced drag increased only marginally when the above constraint was enforced, while the optimal flap settings did change considerably. Small variation of the optimal value in spite of considerable differences in flap deflections indicate that a certain optimal lift distribution can be approximated closely by more than one unique flap deflection pattern.

Wind tunnel experiments

Wind tunnel experiments were performed in the low-speed wind tunnel L2000 at KTH. The tunnel has a 2 by 2 m cross section with corner fillets, and was operated at room temperature and atmospheric pressure. The wing was mounted vertically on the wind tunnel floor. For some experiments, the flexible internal wing structure was replaced by a comparably stiff solid steel beam to investigate flap efficiencies and viscous effects. Aerodynamic loads were measured using a six degree-of-freedom internal balance mounted in the wing root. A splitter plate was placed between the wing and the balance to reduce boundary-layer interactions with the wind tunnel floor. The elastic deformation of the wing was captured using an optical measurement system [18, 19] based on four CCD cameras. Each of the cameras emits infrared flashes and monitors the two-dimensional position of passive reflecting markers attached to the model. By combining the pictures, the three- dimensional position of the markers results from the relative position of the cameras. A sketch of the experimental setup is shown in Figure 5.

Viscous drag Since total drag values from the balance measurements appeared to be higher than the expected drag, an investigation of the impact of the low Reynolds number was performed. For this, the wing was equipped with the rigid beam structure and aerodynamic coefficients were measured at increasing dynamic pressure. The total drag was found to be significantly higher for low Reynolds numbers. When increasing the Reynolds number from 1.5 · 105 to 3.5 · 105, the measured drag coefficient was reduced by almost 50%. The reason for this may be a fairly thick boundary layer at low airspeeds, and possibly regions of laminar flow separation. Due to experimental limitations such as flutter speed and maximum balance loading, testing with the flexible beam was restricted to airspeeds below 30 m/s, corresponding to a Reynolds number of about 3 · 105. This explains the comparatively high total drag values measured during drag optimization investigations. A 14 D. Eller and S. Heinze

Figure 5: Optical deformation measurement using the QualiSys infrared camera system.

Induced drag extraction

To compare experimental measurements with the numerical analysis performed, the induced drag had to be extracted from the measured total drag. Due to viscous effects, however, approaches as described in Barlow et al. [20] based on the measured wing root loads turned out to be insufficient. Therefore, lifting line theory [21] was employed for deriving the induced drag from wing deformation measurements. Since both the two-dimensional lift curve slope cl,α,2d and the corresponding sectional zero-lift angle α0 are variables in the lifting-line formulation, it can be adapted to reproduce the behavior of the wind ◦ tunnel model, for which cl,α,2d = 5.33 and α0 = −3.1 were obtained from experiments with the rigid beam structure. With these modification, the lifting line method accounts for boundary layer effects which lead to a reduction of Induced Drag Reduction with Experimental Evaluation A 15 three-dimensional lift in comparison to the case of inviscid flow. The geometric angles of attack according to

αg(y) = α − α0 + ∆α(y) ∂α ∂α + δT (y) + δL(y) (14) ∂δT ∂δL served as input to the lifting-line computations, where α is the angle of attack of the wing root, α0 is the measured zero-lift angle of attack of the rigid wing and ∆α(y) is the local twist deformation measured by the optical system, see Fig- ure 5. Deflections of leading edge δL and trailing edge flaps δT were accounted for by adjusting the local angle of attack accordingly. The flap efficiencies ∂α ∂α ∂C = · L (15) ∂δ ∂CL ∂δ were obtained from experiments with the rigid beam. With the distribution of local angles of attack, the spanwise distribution of circulation strengths and downwash angles is then calculated using Multhopp’s method [22] with 511 support points. Using the circulation strengths and downwash angles at the support points, lift and induced drag coefficients are obtained. Experimental values for the induced drag given below are all computed from spanwise twist deformations using this approach. Strictly speaking, the lifting line method is only valid for plane, unswept wings as it does not account for spanwise flow. For the small sweep angle of the current wing, it is still considered a valid approximation.

Experimental accuracy The accuracy of the experimental results was estimated based on uncertainties of measured variables. Confidence intervals were calculated for the deflection data from the optical measurements, as well as for the control surface angle. The 95% confidence interval for the uncertainty in the local angle of attack measured by the optical measurement system was found to be below 0.01◦ or ±0.2% of typical wing tip twist deformation. Based on a calibration of the actuator mechanism, the uncertainty of the control surface deflection angle δ was ±0.05◦ within the same confidence interval. This value includes aeroelastic deformations of the control system mechanism for airspeeds up to 35 m/s. Since the geometric angle of attack linearly depends on both angles according to (14), the resulting combined uncertainty is found using partial derivatives to be ±0.032◦. It is likely that measurement errors of this magnitude are small in comparison to the impact of unmodeled physical effects on e.g. control surface efficiencies. A 16 D. Eller and S. Heinze

Results

First, some results of reference experiments are compared with numerical simulations. Then, the drag reductions achieved in computations and experiments are documented along with an investigation on the relative merits of flap configurations with different complexity.

Coefficients and flap efficiency In Table 3, the lift curve slope and zero lift angle are listed for both the rigid and the wind tunnel model with the flexible composite beam. The effect of the flexible structure is to increase both CL,α and α0. The values for the flexible configuration are for an airspeed of 30 m/s and increase further for larger dynamic pressures. Note that, for the angle of attack range of interest, the flexible wing always produces considerably less lift than the rigid configuration due to the pitch down twist of the wing tip.

Case α0 CL,α [1/rad] rigid wing exp. -3.1◦ 5.2 ﬂex. wing exp. +0.3◦ 5.9 rigid wing sim. -4.7◦ 5.5 ﬂex. wing sim. -2.1◦ 6.2

Table 3: Coeﬃcients for rigid and ﬂexible wing.

The numerical model overpredicts the lift curve slope with approximately 0.3/rad, and computes a somewhat smaller zero lift angle. Both differences are expected as the boundary layer, which is currently not included in the simulation, has a decambering effect on the wing, leading to an increase of α0. In this study, the primary interest is in the lift distribution along the span and not the accurate prediction of integral coefficients for low Reynolds number flow. Therefore, in the following, simulation results and experimental cases are compared on the basis of lift coefficient, not angle of attack. This is based on the assumption that, for a given total lift, the spanwise lift distribution can still be compared, which requires that boundary layer thickness distribution does not vary strongly along the span. Considering the very large aspect ratio and minimal sweep angle, the occurrence of significant spanwise flow appears unlikely and the assumption therefore justified. Flap efficiencies were determined experimentally by deflecting one control surface at a time and measuring the difference in lift. The applied deflections Induced Drag Reduction with Experimental Evaluation A 17 were in the order of ±5◦, which was considered to lie within the region of linearity. As expected, the results showed that the leading edge flaps have only very small effect on lift for the rigid wing. On the flexible configuration, the measured effectiveness of leading edge flaps increased by 25%, as a result of the pitch-up twist deformation generated by these flaps. Numerically computed efficiencies for the leading edge flaps agreed well with those measured for the rigid case. However, the simulation model predicted higher than measured leading-edge flap efficiencies CL,δ on the flexible wing. As the twist deformation caused by flap deflections is computed relatively well, the error is likely related to the smaller than predicted lift response of the wing to this twist deformation. Trailing edge efficiencies, in comparison, are consistently overpredicted by the simulation, independent of flexibility, which is usually the case for inviscid flow models [21]. For the flow conditions at hand, it is likely that the fairly thick boundary layer reduces the cambering effect of trailing edge flaps considerably. For the particular configuration considered here, the direct effect of control surface deflections on the lift distribution is small compared to the importance of twist deformation. Due to the very low torsional stiffness, the aerodynamic moments generated by flap deflections lead to quite significant changes in the spanwise twist distribution. It is above all this mechanism which is exploited to achieve favorable lift distributions. For a more torsionally stiff wing, the aerodynamic efficiency, in particular that of the trailing edge flaps, would gain much in importance.

Optimization Numerical optimization is performed for lift coefficients in the range 0.2 to 0.8, and the results are compared with a baseline configuration for which only the angle of attack is changed to obtain the target lift coefficient. Both the baseline and the cases with optimized flap settings were tested in the wind tunnel. In order to achieve identical lift coefficients, simulations were performed at root incidence angles which were between 1.6◦ and 2.4◦ smaller than those run in the wind tunnel. Results are presented as diagram of induced drag over lift coefficient. For the case presented in Figure 6, wing loading remains constant over the CL range, meaning that the lower CL values correspond to higher dynamic pressure q∞. This type of analysis is meant to model operation at different altitudes and speeds for constant aircraft weight. Especially for relatively low lift coefficients, the experimentally determined induced drag estimations fit very well with the numerical predictions both for the baseline and the optimized flight conditions. For all tested flight conditions, an improvement could be achieved. For high lift coefficients, however, the A 18 D. Eller and S. Heinze

exp. baseline 0.014 sim. baseline exp. optimized 0.012 sim. optimized

0.01

0.008

0.006

Induced drag coefficient 0.004

0.002

0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Lift coefficient

Figure 6: CDi over CL for varying dynamic pressure q∞ with constant wing loading of 150 N/m2. simulation predicts slightly larger savings than measured. From the variation of the induced drag for the clean configuration without flap deflections it can be concluded that there are operating conditions for which the wing in question is already close to optimal. This is representative for an actual aircraft design problem, where an elastic wing would likely be designed with an aerodynamic twist distribution which yields optimal performance for a certain design point (dynamic pressure and CL). Nevertheless, significant drag reductions can be achieved at other operating points by employing control surface deflections. For the aeroelastic configuration considered here, the most significant drag reduction can be obtained at low CL values and high dynamic pressures, conditions at which low induced drag would be usually expected. The numerical −4 optimization yields a maximum saving of about ∆CDi = 25·10 at CL = 0.2 and 40 m/s. The reason that relatively large gains can be achieved for this condition is the unfavorable lift distribution of the wing without flap deflections. The distribution is characterized by a downward force in the outboard section with large negative twist angles, and a slightly larger upward force on the inboard part of the wing. This S-shaped lift distribution generates a considerable amount of induced drag at low CL values. Even for moderate lift values, the lift distribution is still unfavorable in terms of induced drag, showing an approx- Induced Drag Reduction with Experimental Evaluation A 19 imately triangular shape where the outboard 20% of the wing contribute very little to lift. By modifying the spanwise twist distribution using control surface deflections, a lift distribution with much less drag can be reached. Figure 7 shows the lift distribution in terms of local lift coefficient over the semispan for a specific case with CL = 0.2 and u∞ = 30 m/s. The sharp peaks seen in the computed lift distribution at certain positions result from the geometry of the wind tunnel model. Since the rigid aerodynamic sections are only attached in points on the internal structural beam, the wing twist angle does not vary smoothly along the span. Discrete jumps in twist angle between the wing segments cause the irregular circulation distribution, which, due to the small magnitude of the peaks, has negligible influence on induced drag. The experimentally obtained distribution does not resolve this small-scale effect. Lift distributions for optimized flap settings are very close to the elliptic spanwise distribution of circulation, showing the effectiveness of the numerical optimization.

exp. baseline sim. baseline 0.4 exp. optimized sim. optimized

0.3

0.2

0.1 Section lift coefficient

−0.1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Span coordinate [m]

Figure 7: Spanwise lift distributions.

The experimental lift distributions presented in Figure 7 result from lifting- line analysis using the measured twist angles and control surface settings. Both angles are aﬀected by measurement inaccuracies, as described above, leading to a maximum error of less than ±3% in local lift coeﬃcients. A 20 D. Eller and S. Heinze

Flap configurations The results shown in Figures 6 and 7 were obtained for a configuration with 20 individual control surfaces and the angle of attack as design variables. While such a setup enables a fairly accurate control of the beam deformation and the distribution of camber, it is probably too complex for an aircraft wing. For the more realistic, simplified flap configurations shown in Figure 4, the achievable drag reductions are slightly smaller. In Table 4, the possible reduction of induced drag ∆CDi is given along with the corresponding value of Oswald’s span efficiency

C2 e = L , (16) πΛCDi A span eﬃciency of 1.0 corresponds to the induced drag of the elliptic lift distribution. The listed values were computed for the case with CL = 0.15 at u∞ = 46 m/s, where the possible drag savings are relatively large. Wing root incidence angles α given in Table 4 refer to values computed by the optimization procedure for the numerical model. Optimal control surface deﬂections for this particular case are listed in Table 5.

n ∆CDi e α baseline 0.0 0.256 2.28◦ 3 −17.9 · 10−4 0.909 3.80◦ 6 −18.2 · 10−4 0.957 1.26◦ 10 −18.4 · 10−4 0.979 1.24◦ 20 −18.4 · 10−4 0.986 1.17◦

Table 4: Drag reduction and span efficiency obtained for different flap configurations.

Table 4 suggests that three trailing edge control surfaces alone are sufficient to achieve a significant reduction in induced drag. However, this is only true in connection with the corresponding wing root incidence angle of 3.8◦ , which is one of the design variables. By using leading edge flaps, slightly better induced drag reductions can be obtained without the need of an increased root incidence angle. This may be relevant for configurations where a reduced range of operating angles of attack entails advantages in parasitic drag. The advantage of using leading edge flaps for torsionally flexible wings lies in their ability to generate considerable pitch-up twist moment and a small positive aerodynamic lift increment. Naturally, trailing edge surfaces are much more Induced Drag Reduction with Experimental Evaluation A 21

Sections 1-2 3-4 5-6 7-8 9-10 Leading Edge n = 6 10.0 10.0 10.0 10.0 10.0 n = 10 10.0 10.0 10.0 10.0 9.1 Trailing Edge n = 3 −3.8 −3.8 −6.3 −6.3 −10.0 n = 6 −8.4 −8.4 −7.5 −7.5 −7.7 n = 10 −8.8 −8.0 −7.7 −7.4 −8.0

Table 5: Optimal flap settings for the case CL = 0.15 at u∞ = 46 m/s. efficient in generating additional lift, however, at the same time, they always produce a pitch-down twist moment, causing a deformation of the complete wing counteracting the lift increment. Therefore, leading edge flaps can be more effectively used to control wing twist deformation. Drag reduction efforts in aircraft design will always aim at reducing total drag of the complete trimmed configuration, which may depend in a complex manner on angle of attack and flap deflections. The potential benefits of leading edge flaps described above must hence also be judged by the effect they most likely have on viscous drag components.

Computational aspects When formulating the drag minimization problem as a linear least-squares problem according to (9)-(10), the computational cost is moderate. For 6 design variables, a single solution to the optimization problem is computed in about 15 minutes on a desktop computer with 1.4 GHz Athlon processor. Using pressure integration and nonlinear programming, mesh requirements and the large number of finite-difference gradient calculations lead to at least tenfold increased computational effort. With mesh resolutions sufficient for accuracy, computation times become unacceptable.

Conclusions

The investigation demonstrated that the induced drag of an elastic wing configuration can be reduced significantly by means of conventional leading and trailing edge control surfaces. The use of leading edge control surfaces can be beneficial if the optimal twist distribution requires that large sectional pitch moments are generated. Furthermore, the use of more than six control surfaces does A 22 D. Eller and S. Heinze not appear to pay off, unless the marginal additional savings really compensate for the increased cost and complexity. If savings of relevant magnitude are possible in the first place depends primarily on the quality of the lift distribution and the flexibility of the wing. Relatively rigid wings will show less variability of their spanwise loading under different flight conditions, while highly flexible ones will likely operate with very unfavorable lift distributions in parts of the flight envelope. These are the configurations where substantial drag reductions can be achieved by means of control surfaces. The use of potential flow methods for a induced drag reduction problem at Reynolds numbers below 106 is possible, but difficult. Boundary layer effects have a significant impact on chordwise pressure distribution and control surface efficiencies, so that potential flow results may be hard to interpret when experimental comparisons are not available. The inclusion of a boundary layer model in the numerical method is expected to improve the prediction of control surface efficiencies in particular.

Acknowledgments

Construction of the wind tunnel model, definition of the structural analysis model as well as current and upcoming experimental activities are financed by the European Union under the Fifth Research Framework, through the project Active Aeroelastic Aircraft Structures, project number GRD-1-2001-40122. The first author’s work, comprising aerodynamic modeling, optimization and computational studies, are financed by the Swedish National Program for Aeronautics Research (NFFP).

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[5] P. S. Zink, D. N. Mavris, and D. E. Raveh. Maneuver Trim Optimiza- tion Techniques for Active Aeroelastic Wings. AIAA Journal of Aircraft, 38(6):1139–1146, 2001.

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[8] S. Kuzmina, F. Ishmuratov, V. Kuzmin, and Y. Sviridenko. Estimation of Flying Vehicle Induced Drag Changing Due to Deformation of Lifting Surfaces. In CEAS/AIAA/NVvL International Forum on Aeroelasticity and Structural Dynamics, Amsterdam, The Netherlands, June 2003.

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Drag Minimization of an Active Flexible Wing with Multiple Control Surfaces

Marianne Jacobsen and Sebastian Heinze Royal Institute of Technology SE-100 44 Stockholm, Sweden

Abstract In this study, the performance of a highly flexible wing with multiple control surfaces is optimized for different flight conditions. A wind tunnel model with 20 independent control surfaces is used as the test object. The problem is posed as a nonlinear optimization problem with the measured drag as the objective. Constraints include keeping the lift constant, as well as having bounds on the control surface deflections. No computational estimation of the drag is performed. Instead, experimental measurements of the drag are used to perform the optimization in real time using a generating set search method (GSS). The algorithm makes the wing adapt to the current flight condition to improve performance and the experimental evaluation of the method shows that the drag is significantly reduced within a wide range of flight conditions. A study on the number of control surfaces needed is also included, and the results show that, if chosen properly, only three control surfaces distributed over the wing are needed to reduce drag significantly.

Introduction

Minimizing drag of an aircraft is of great importance, both from an environ- mental and an economical perspective. Much is already performed during the design stage when the shape of the aircraft is determined to minimize drag for a specified flight condition, or for multiple design points that compromise the overall performance [1]. Today, many aircraft operate in a wide range of flight conditions. Examples of such aircraft are long-range commercial transports that are subject to a considerable change in fuel weight. Other examples are surveillance UAVs, which have a very high fuel weight ratio, or vehicles, which operate at different speeds or altitudes. Much could possibly be gained if the aircraft could be optimized for a wider range of flight conditions [2].

B 1 B 2 M. Jacobsen and S. Heinze

Real time drag minimization by changing the geometry during flight has been described in Refs. [3, 4, 5, 6]. It is also stated that the economic savings can be as much as $140,000 each year for each transport aircraft if the drag is reduced by as little as 1%. The method developed in these studies was tested on a modified L-1011 [4], and the results show that the drag could be decreased approximately 1% when using only the outboard ailerons. In real flight experiments, the speed can be used as an indicator for drag, and the flight tests performed with the L-1011 show that this is in fact possible and that it gives satisfying results. The objective with this study is to exploit multiple control surfaces and wing flexibility to minimize drag for different flight conditions. A study by Eller and Heinze [7] shows that induced drag can be minimized in this way, but in the present study the total drag is considered instead. The idea is to use optimization to minimize drag for a given flight condition using the control surfaces. The objective function is the drag coefficient, which is here evaluated by performing drag measurements for the current geometry. Using measurements as the objective function puts certain demands on the method used to solve the problem since this will introduce difficulties such as noise in the signals, hysteresis, and problems with repeatability in the function evaluations. A method, which deals with these difficulties, has been used in a previous study by Jacobsen [8] with promising results. The method is known as a generating set search method (GSS) [9], which solves an optimization problem without explicit use of gradients. This is a desirable property when real time measurements are performed since derivatives of a signal with noise can be difficult both to define and evaluate. In this study, the same approach is applied to a highly flexible wing with 20 independent control surfaces. An investigation on the number of control surfaces needed to obtain significant drag reduction is also included.

Optimization

Minimizing drag as a function of control surface deﬂections can be posed as an optimization problem in the form

min CD(α, δ) α, δ (1a)

subject to CL(α, δ) = CLset, (1b)

δl ≤ δ ≤ δu. (1c)

Here, CD denotes the drag coefficient and CL is the lift coefficient. The angle of attack is denoted α. The vector δ contains the control surface deflections that may vary between lower and upper bounds, here denoted δl and δu respectively. Drag Minimization of an Active Flexible Wing B 3

The equality constraint corresponds to keeping the lift constant at the desired lift coefficient CLset. This formulation is general since no assumption is made on how the lift and drag coefficients depend on the control surfaces. Due to the experimental facilities, the angle of attack is treated as a parameter rather than a variable since changing the angle of attack in the wind tunnel takes much time compared to deflecting the control surfaces. To simplify the optimization problem the lift coefficient is assumed to be a linear function of the control surface deflections. Linear superposition of control surface deflections has been studied by Heeg [10], and it was concluded that lift coefficient results are generally captured reliably by linear superposition. A linear constraint simplifies the optimization problem considerably. The solution must lie in the nullspace of the constraint, which makes the space of possible solutions smaller. If the lift coefficient at zero control surface deflections is denoted CL0, and if the vector of first derivatives of CL with respect to δ is denoted CLδ, the optimization problem can instead be written as

min CD(δ) δ (2a) T subject to CLδδ = CLset − CL0, (2b) δl ≤ δ ≤ δu. (2c)

The drag is measured in real time in this study so that no simulation model is required. However, the measured signal contains noise, and diﬃculties such as hysteresis and lack of repeatability in the function evaluations are introduced in this way. This makes it diﬃcult, if not impossible, to use a gradient based method to solve (2). Instead, a GSS method is used that only requires function evaluations. A short description of the GSS method follows, but more detailed information is found in Lewis et al. [11] and Jacobsen [8].

Generating set search As the name implies, a GSS method generates a set of search directions at each iteration. The trial points δtest are chosen as

δtest = δk ± ∆kdk, (3) where δk is the iterate at iteration k, ∆k is the step length and dk is the search direction. The search directions are chosen in the nullspace of the linear constraints, and the inequality constraints are accounted for by conforming the search directions to the nearby boundaries. Nearby is here deﬁned as closer to the current iterate than a distance ∆k. This is illustrated in Figure 1, where the feasible region is within the triangular region (the nonshaded region in B 4 M. Jacobsen and S. Heinze

δk

|∆k|

Figure 1: The search directions adapted to the local geometry.

Figure 1). The asterisks represent the trial points, which are here chosen taking the two nearby inequality constraints into account. If no constraints are nearby, the search directions would be chosen as the coordinate directions, but in this case they are chosen parallel and perpendicular to the identified inequality constraints. The trial points in the directions perpendicular to the inequality constraints are, however, not at a distance ∆k from the current iterate since that makes the trial points infeasible. Choosing the search directions in this way makes the search directions conform to the feasible region. The details of the computation of the search directions for the method used in this study are found in Refs. [8, 11]. Once the trial points have been determined, the drag coefficient is evaluated both at the current iterate δk and at the trial points. The value of the objective function is denoted CD(δk). To accept a new iterate the sufficient decrease condition

CD(δtest) < CD(δk) − ρ(∆k), (4) Drag Minimization of an Active Flexible Wing B 5

must be satisﬁed. Here, ρ(∆k) is the nonnegative function

ρ(∆) = µ∆2, (5) where µ is a constant chosen to reflect the magnitude of CD, ∆, and the signal noise. Here, µ = 3 · 10−4 deg−2 is used. If the sufficient decrease condition is not satisfied by any of the trial points, the step length is reduced. The contraction ratio is set to 1/2. The GSS iterations are terminated once the criteria ∆k < ∆tol (6) is satisfied. The first iterate δ0 of the GSS method must be feasible. If it is infeasible with respect to any of the constraints, a new first iterate δ is chosen as the point in the feasible region closest to δ0. This gives a Euclidean projection according to

1 2 min k δ − δ0 k δ 2 (7a) T subject to CLδδ = CLset − CL0, (7b) δl ≤ δ ≤ δu, (7c) where k · k denotes the Euclidean norm. An important property of any optimization algorithm is convergence. The GSS method for linearly constrained problems can be shown to achieve convergence without the explicit use of gradients [12]. The performance of the GSS method degrades quickly with an increasing number of variables. Despite this, the GSS has some advantages since it can be used on almost any type of function, even when, like in this case, derivatives are not available.

Experimental setup

Wind tunnel tests are performed in the low speed wind tunnel L2000 at Kungliga Tekniska Hogskolan¨ (KTH). The wind tunnel cross section is 2 × 2 meters with corner fillets, and it is operated at room temperature and atmospheric pressure. The test object in this study is a high aspect ratio wing with a semispan of 1.6 m and a chord of 0.16 m, shown in Figure 2. It was designed and built for aeroservoelastic investigations as part of the European Union project Active Aeroelastic Aircraft Structures (3AS) [13]. The wind tunnel model has been used in previous studies on drag reduction, see Eller and Heinze [7]. The load carrying structure is a beam of carbon-fiber composite, and 10 aerodynamic sections are rigidly clamped to the beam. Each section has a leading edge flap and a trailing edge flap, which gives the wind tunnel model 20 B 6 M. Jacobsen and S. Heinze

Figure 2: Illustration of the experimental setup. independent control surfaces. The control surfaces can be actuated during the tests in the wind tunnel using electromechanical servo actuators located inside the sections to minimize flow disturbances. The wing is mounted vertically on the wind tunnel floor with a six-component internal balance at the wing root. To reduce the boundary layer interaction between the model and the wind tunnel floor, a splitter plate is placed between the balance and the model.

Testing procedures

The GSS method is implemented in the graphical programming software LAB- VIEW [14]. Although the wind tunnel model has 20 independent control surfaces, the number of design variables n can be reduced by deflecting groups of control surfaces together. With, for example, two variables, all control surfaces on the leading edge or on the trailing edge are deflected equally. A sketch of the different configurations investigated in this study is shown in Figure 3. It was found that increasing the number of variables decreases the quality of the measured data, since the control authority of individually operated control surfaces Drag Minimization of an Active Flexible Wing B 7

Figure 3: Control surface conﬁgurations.

is too low to change CL significantly, leading to unreliable data for CLδ. For this reason, control surfaces are operated in groups of at least two, leading to the configuration with ten variables. The optimization is performed at a fixed flight condition given by CLset and the free-stream velocity u∞. Before the optimization is started the slopes CLδ are determined. The lift curve given by CL0 and CLα, the derivative of CL with respect to α, is also measured. This is used to find the angle of attack that gives the desired lift for δ = 0. This control surface setting is used as a starting point for the optimization together with the adjusted angle of attack. The angle of attack is kept constant during the optimization.

Results

The drag minimization algorithm is experimentally verified at different flight conditions. The effect of the number of control surfaces is also investigated, as well as the effect of different Reynolds numbers. A short discussion on the impact of flexibility is also included.

Varying flight conditions The GSS method is applied to the wind tunnel model for different values of CL. The wind tunnel speed is kept constant at u∞ = 25 m/s. The angle of attack is adjusted in order to obtain the desired lift at zero control surface deflections, referred to as the baseline configuration. The number of design variables is n = 6, see Figure 3 for the layout of the variables. Using 6 control surfaces, instead of the possible 10, is mainly to decrease the duration of the experiment while keeping the level of complexity reasonably high. For each value of CL, or α, the drag is first measured for the baseline configuration and the optimization is then performed. The optimized drag coefficient, as well as the drag for the baseline configuration, is shown in Figure 4. The drag can be B 8 M. Jacobsen and S. Heinze

0.1 baseline optimized

0.08 D

C 0.06

0.04

0.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 C L

Figure 4: CD over CL for constant airspeed. significantly reduced for the higher lift coefficients, but the drag reduction is approximately 10% even at CL = 0.1. At the higher lift coefficients in Figure 4, there is a small difference in lift of the plotted baseline drag and the optimized drag. This is due to the assumption that the lift is a linear function of control surface deflections, and for some combinations of control surface deflections, linear superposition does not give accurate results, as shown in Heeg [10]. The experimental results in Figure 4 can be compared to the results in a previous study by Eller and Heinze [7], which uses the same wind tunnel model but focuses on minimizing the induced drag by making the lift distribution elliptic [15]. It was shown that the induced drag could be reduced significantly by using the optimal control surface settings. The maximum savings of the −4 induced drag was ∆CDi = 25 × 10 at CL = 0.2. On the other hand, at CL = 0.4 the savings were nearly negligible. The induced drag is much smaller than the total drag in this case, which also makes the possible drag reduction much smaller when only induced drag is considered. Since the total drag is minimized in this study, the small improvements that can be made by making the lift distribution elliptic are negligible. The large total drag measured here indicates a complex flow situation. The flow physics may be difficult to model accurately with computational methods. Despite this, the GSS method used for drag minimization works well and is shown to reduce drag considerably.

The optimal control surface deﬂections as a function of CL are shown in Figure 5. The control surface settings correspond to the six variables distributed across the span as shown in Figure 3. The ﬁrst variables are located at the wing Drag Minimization of an Active Flexible Wing B 9

LE1 (root) 4 TE1 LE2 TE2 LE3 TE3 (tip) 2

−2 Control surface settings [deg] −4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 C L

Figure 5: Optimal control surface settings at different CL. root on the leading edge (LE1) and on the trailing edge (TE1). The bounds ◦ ◦ on the control surface deflections are chosen as −4.6 ≤ δi ≤ 4.6 . It was found that the deflection pattern of the trailing edge control surfaces are more coherent than the leading edge control surfaces that vary more and do not show a specific tendency.

Control surface configurations To study the impact of the number of control surfaces on the drag minimization, the optimization is performed for an increasing number of variables. For CL = 0.5, the optimization is started with two variables only, and terminated after a few iterations. The optimal control surface setting is then kept, but instead four design variables are used. This increase of dimensions leads to further drag reduction and when the optimum is found the number of variables is increased to n = 10. The configurations used for the different cases are the ones shown in Figure 3. The change in drag coefficient and the change in lift during the optimization is shown in Figure 6, which also shows lift and drag of the corresponding case with n = 10 variables from the beginning. The changes in lift and drag in Figure 6 are plotted versus the number of function evaluations, since that is the time consuming part of the experimental optimization. The results in Figure 6 show that it is more time efficient to increase the number of variables in steps instead of using the full complexity from the beginning. The same drag reduction is obtained, but with fewer function evaluations. Using 10 control surfaces in the optimization is rather time consuming. The B 10 M. Jacobsen and S. Heinze

←n=2 0.066 n=10 n=2, 4 and 10

D n=4 C 0.06 n=10 0.054 0 50 100 150 200 250 300

0.6

L ←n=2 C 0.5

n=4 n=10

0.4 0 50 100 150 200 250 300 Number of function evaluations

Figure 6: Lift and drag coefficient for increasing number of control surfaces. duration of the experiment is approximately 25 minutes, but it depends heavily on the number of iterations needed to converge. It may also be expensive to equip an aircraft with many control surfaces due to, for example, manufacturing reasons [2]. To investigate if the number of variables can be reduced by join- ing groups of control surfaces without increasing the optimal drag coefficient substantially, the control surface settings during the optimization with n = 10 variables is studied, as shown in Figure 7. The optimal control surface setting is the final values at iteration 25. Using this information it seems unnecessary to use both TE4 and TE5 as two different control surfaces. The leading edge control surfaces are rather close to zero at all times, but to keep some of the complexity on the leading edge, the control surfaces can, for example, be divided into two groups. The three innermost trailing edge control surfaces show different tendencies, which implies that they should be kept to maintain the drag reduction possibility. This grouping of control surfaces decreases the number of variables from n = 10 to n = 6. If even less variables should be used, a compromise between the loss of possible drag reduction and the decrease of complexity is made. A planform, which uses only three control surfaces can also be constructed using the information in Figure 7. The two new control surface configurations are shown in Figure 8. The lift and drag coefficient during optimization with CLset = 0.5 from the baseline configuration using the two new configurations are shown in Figure 9. The 10 variable case is also shown for comparison. The optimal drag coefficient is slightly less for the configuration with ten control surfaces compared to the two new configurations. The difference between the configurations is small compared to the total Drag Minimization of an Active Flexible Wing B 11

LE1 (root) TE1 4 LE2 TE2 LE3 TE3 2 LE4 TE4 LE5 TE5 (tip) 0

−2 Control surface settings [deg] −4

0 5 10 15 20 25 Iteration

Figure 7: Control surface settings during optimization with n = 10.

Figure 8: Control surface layout.

reduction in CD and the results, therefore, show that it is possible to reduce the complexity quite signiﬁcantly and still obtain large drag reductions. Even the three variable conﬁguration shows almost as much decrease in CD as the other two cases.

Reynolds number effects The GSS method has so far been shown to work well in a rather wide range of flight conditions as well as for different number of control surfaces. The Reynolds number of the preceding experiments has been low, Re = 2.7 · 105, which corresponds to a speed of u∞ = 25 m/s. At airspeeds above 30 m/s, flow separation induces fairly violent aeroelastic oscillations for some control surface deflections on the wind tunnel model. Hence, higher Reynolds numbers are not possible to use in the experiments. In Figure 10, the drag optimization is shown for different speeds. The different Reynolds numbers make the drag coefficient vary somewhat. The decrease in drag for the two higher speeds in Figure 10 is substantial, but when the speed is reduced to u∞ = 15 m/s the algorithm has difficulties obtaining B 12 M. Jacobsen and S. Heinze

n=10 0.065 n=6 n=3 D

C 0.06

0.055 0 5 10 15 20 25

0.6 L

C 0.5

0.4 0 5 10 15 20 25 Iteration

Figure 9: Lift and drag coefficient for different configurations.

any signiﬁcant reduction in CD, and the drag is almost constant. At this speed the forces are very small, which also makes the change in drag very small due to a change in geometry. Hence, the small changes in drag are too diﬃcult to detect in the noise.

Aeroelastic effects All wings are flexible to some extent. To save weight, the tendency is to decrease structural weight of aircraft today, which results in even more flexible wings. The aircraft is designed for a certain flight condition that gives low drag. At other operating points, the flexibility can lead to deformations that increase the drag considerably. Distributed control surfaces along the span can be used to control the deformation and indirectly also the drag. The potential of using redundant control surfaces to decrease drag is, therefore, assumed to be larger for flexible wings compared to fairly rigid wings. Flexibility of wings is also difficult to model and take into account when designing aircraft. The method used in this study solves the problem without using explicit information about the deformation or lift distribution. The deformation, both bending and twist, of the wind tunnel model is substantial at higher speeds. In Figure 11, a multiple exposure photograph from the wind tunnel experiments is shown. Three different states are seen, the ’wind off’ state is to demonstrate the rather large deformations the wing undergoes when the wind is turned on. There is also a clearly visible change in bending deformation between the baseline configuration and the optimized. If the wing Drag Minimization of an Active Flexible Wing B 13

u=15 m/s 0.065 u=20 m/s u=25 m/s

D 0.06 C

0.055

0 2 4 6 8 10 12

0.6 L

C 0.5

0.4 0 2 4 6 8 10 12 Iteration

Figure 10: Lift and drag coefficient for different flow velocities. tip is studied, the optimized state can be seen to have a less nose-down twist deformation compared to the baseline.

Conclusions

The increase in performance of a wing with 20 independent control surfaces has been investigated. The total drag of the wind tunnel model is reduced significantly for different flight conditions, a decrease of approximately 10-15% can be achieved when using 6 groups of control surfaces. The method used in this study is simple. It does not require any knowledge about CFD methods or even the physics involved. The objective is to decrease drag for any flight condition, even for operating points far from the design point. For aircraft with flexible wings, aeroelasticity must be considered when designing the wing. To obtain reliable results from an aeroelastic analysis, the structure as well as the mass distribution must be modeled precisely. The structural model should then be connected with the aerodynamic model. This is not simple and it may take much time to obtain reliable results. The GSS method in this study does not use any models of the aircraft. The measurements are performed on the real aircraft, which means that aeroelastic effects are present and the method indirectly take this information into account while minimizing the drag. An aircraft is commonly only optimal for a few design points where the aircraft is assumed to be operated. The task of using multiple design points B 14 M. Jacobsen and S. Heinze

Figure 11: Diﬀerence in deformation.

is difficult. The method used in this study can not be used during the design stage, but it is instead intended to increase performance of the finished aircraft. If the aircraft is operated at many different flight conditions, the increase in performance at some conditions far from the design point may be substantial. Redundant control surfaces are available during cruise on almost all transport aircraft today. To fully use the potential of this method for drag reduction during flight, there may be more efficient ways to place the control surfaces, but the possibility exists already.

Acknowledgments

The design and manufacturing of the wind tunnel model was ﬁnanced by the European Union under the Fifth Research Framework, through the project Active Aeroelastic Aircraft Structures, project number GRD-1-2001-40122. Drag Minimization of an Active Flexible Wing B 15

References

[1] E. Stanewsky. Aerodynamic Beneﬁts of Adaptive Wing Technology. Aerospace Science and Technology, 4(7):439–452, 2000.

[2] E. Stanewsky. Adaptive Wing and Flow Control Technology. Progress in Aerospace Sciences, 37(7):583–667, 2001.

[3] G. Gilyard. Development of a Real-Time Transport Performance Optimiza- tion Methodology. Technical Report TM 4730, NASA, 1996.

[4] G. B. Gilyard, J. Georgie, and J. S. Barnicki. Flight Test of an Adaptive Con- ﬁguration Optimization System for Transport Aircraft. Technical Report TM-1999-206569, NASA, 1999.

[5] M. Espana˜ and G. Gilyard. Direct Adaptive Performance Optimization of Subsonic Transports: A Periodic Perturbation Technique. Technical Report TM 4676, NASA, 1995.

[6] G. Gilyard. In-Flight Transport Performance Optimization: An Experi- mental Flight Research Program and an Operational Scenario. Technical Report TM-1997-206229, NASA, 1997.

[7] D. Eller and S. Heinze. An Approach to Induced Drag Reduction with Experimental Evaluation. AIAA Journal of Aircraft, 42(6):1478–1485, 2005.

[8] M. Jacobsen. Real Time Drag Minimization Using Redundant Control Surfaces. Aerospace Science and Technology, 10(7):574–580, 2006.

[9] T. G. Kolda, R. M. Lewis, and V. J. Torczon. Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods. SIAM Review, 45(3):385–482, 2003.

[10] J. Heeg. Control Surface Interaction Eﬀects of the Active Aeroelastic Wing Wind Tunnel Model. In 47th AIAA/ASME/ASCE/AHS/ASC Structures, Struc- tural Dynamics and Materials Conference, Newport, Rhode Island, USA, May 2006.

[11] R. M. Lewis, A. Shepherd, and V. J. Torczon. Implementing Generating Set Search Methods for Linearly Constrained Minimization. Technical Report WM-CS-2005-01, Department of Computer Science, College of William and Mary, Williamsburg, Virginia, 2005. To appear in SIAM Journal on Scientiﬁc Computing. B 16 M. Jacobsen and S. Heinze

[12] T. G. Kolda, R. M. Lewis, and V. J. Torczon. Stationarity Results for Gen- erating Set Search for Linearly Constrained Optimization. SIAM Journal on Optimization, 17(4):943–968, 2006.

[13] J. Schweiger and A. Suleman. The European Research Project Active Aeroe- lastic Aircraft Structures. In CEAS/AIAA/NVvL International Forum on Aeroelasticity and Structural Dynamics, Amsterdam, The Netherlands, June 2003.

[14] G. W. Johnson. LabVIEW Graphical Programming. McGraw-Hill, 1994.

[15] J. D. Anderson. Fundamentals of Aerodynamics. McGraw-Hill Book Com- pany, Inc, ISBN 0-07-001679-8, 2nd edition, 1991. Paper C

Analysis and Wind Tunnel Testing of a Piezoelectric Tab for Aeroelastic Control Applications

Sebastian Heinze Royal Institute of Technology SE-100 44 Stockholm, Sweden Moti Karpel Technion - Israel Institute of Technology Haifa 32000, Israel

Abstract

A concept for exploitation of a piezo-electric actuator by using aeroelastic amplification is presented. The approach is to use the actuator for excitation of a tab that occupies the rear 25% of a free-floating trailing edge flap. A flexible high aspect ratio wing wind tunnel model is used as a test case. Wind tunnel experiments were performed to determine frequency response functions for validation of the numerical model used for the control law design. The aeroservoelastic model is based on state- space equations of motion that accept piezoelectric voltage commands. Control laws are derived for gust alleviation with flutter and control authority constraints, and numerical results that demonstrate a significant reduction in the structural response are presented. Possible applications and feasibility of the concept are discussed.

Introduction

Piezoelectric materials have been in the focus of aeronautical research for many years [1, 2, 3]. Especially high bandwidth and small size are beneﬁcial properties that allow for excitation that is not possible with conventional electric or hydraulic actuators. Recent applications of piezo-electric materials are found in aeroelastic and vibration control [4, 5, 6, 7] as well as in the excitation of structures for the purpose of e.g. parameter identiﬁcation [8]. Despite a wide range of applications, however, these materials are mostly used in research, rather than

C 1 C 2 S. Heinze and M. Karpel in real aircraft structures. Two of the main reasons for this are the relatively low actuator stroke and allowable strains. The focus of this study is on the efficient exploitation of the advantages of piezo-electric materials, i.e. small size and high bandwidth, while placing the actuator in a low-strain area and trying to compensate for the low stroke by aeroelastic amplification. This is done by using the piezoelectric material for excitation of a tab connected to a floating control surface, rather than controlling the surface itself. Aerodynamic amplification using trailing edge tabs is widely used in static applications, such as elevator trim tabs, and has been proven very efficient. In this study, the dynamic aeroelastic response of such a tab under piezoelectric actuation, when the flap is floating and does not have other actuators, is investigated. The study is performed within the project Active Aeroelastic Aircraft Structures (3AS) [9], which is funded by the European Union under the Fifth Framework Programme. As a test case, a high aspect ratio wing (HARW) wind tunnel model built within the project is considered. The wind tunnel model is a generic model of a high-altitude long-endurance surveillance aircraft wing that has been developed for a drag minimization study [10] and was modified for integration of the piezo-electric actuator.

Experimental setup

Experiments were performed in the low-speed wind tunnel L2000 at the Royal Institute of Technology (KTH) at airspeeds of 25 m/s at atmospheric pressure and room temperature. The 1.6 m semi-span model is mounted vertically in the wind tunnel floor. In the current configuration, the wing consists of ten aerodynamic sections, of which one is equipped with the piezo-electric tab, as shown in Figure 1. The stiff sections are clamped to a slender load-carrying flexible beam in a way that minimizes their contribution of stiffness to the structure. A cross-sectional view of the section containing the piezoelectric actuator is shown in Figure 2. The actuator mechanism is mounted to a frame within the floating trailing edge flap. Voltage excitation of the bimorph PZT actuator results in vertical deflection at the actuator tip, located half way between the flap hinge and the tab axis, leading to angular deflection of the tab, as shown in the sketch. The QP25N PZT actuator [11] of ACX was used in this study. The device is restricted to the maximal input of 100 Volt which yields, according to the manufacturer, a maximal peak-to-peak free displacement of 1.42 mm or the maximal zero-to-peak blocked force of 0.32 N, from which the piezoelectric and stiffness properties are extracted below. Due to mechanical restrictions, the Piezoelectric Tab for Aeroelastic Control Applications C 3

Figure 1: Experimental setup. The accelerometers are located inside the model.

Figure 2: Cross sectional view of the section with a piezo-actuated tab. C 4 S. Heinze and M. Karpel maximum deflection of the control surface is less than 3◦ . It should however be noted that the focus is on amplification of this deflection by the floating flap, rather than on implementing an actuator mechanism featuring large deflections. The objective of the experimental investigations was to derive frequency response functions (FRFs) relating dynamic tab deflections to structural response. The piezoactuator was therefore excited with a sine wave generator at several frequencies, and the structural response was measured using accelerometers located inside the wing as shown in Figures 1 and 2. The main purpose of having accelerometer number 4 is to extract the flap motion from accelerometer number 3. The tab deflection could not be measured directly during the experiments. Therefore, strain gauges were attached to the piezo actuator, and the strain in the actuator skin was considered to be a measure for the actuator tip displacement, which could be related to the tab deflection. The entire setup was calibrated statically, and it was found that due to fairly low inertial and aerodynamic loads on the tab, this calibration was sufficient for the frequency range considered here (up to 20 Hz). Accelerations and piezoelement strain were measured simultaneously to guarantee that there was no time delay between the signals. Experiments were performed with both the floating flap and with a locked flap to investigate the amplifying effect of the floating configuration. Note that the floating flap is not perfectly free due to cables connecting the PZT element to the rigid aerodynamic section.

Numerical modeling

Nastran A numerical model of the wing was developed in MSC/Nastran. Beam elements were used for modeling of the load-carrying spar. Structural properties were derived from material data and adjusted to both static experiments and vibration tests of the beam without the sections applied. Mass properties of the sections were derived from measurements and known material properties. Since each of the sections is attached to the beam in one point only without stiffness contribution to the beam, the sections were simply modeled as concentrated mass elements. The structural properties were finally validated by vibration tests with all sections applied to the clamped wing. The flap was given some stiffness and damping in the rotational degree of freedom to account for cables connected to the piezo actuator. Even though the rotational stiffness is fairly low, it was found necessary for correct modeling of the flap dynamics. Throughout this paper, this configuration will be referred to as the stiffened configuration, as opposed to the free-floating case referring to the perfectly free flap. The flap and tab kinematics were described Piezoelectric Tab for Aeroelastic Control Applications C 5 as multipoint constraints such that the flap deflection relative to the wing and the tab deflection relative to the flap appear as independent structural degrees of freedom. Since MSC/Nastran does not facilitate for piezoelectric elements, the numerical model validation by comparisons with measures FRFs was performed with the tab deflection degree of freedom loaded with a large inertia of 104 kgm2. In this way, the acceleration response to a sinusoidal moment excitation at this point with the amplitude of 104 Nm is equal to the displacement FRF to tab deflection excitation. The disadvantage of this modeling is that tab dynamics are ignored, which did not cause a significant error in the frequency range of interest of our case. Flutter analyses were performed in the validation process with a locked tab. A different approach to the tab modeling was taken in the generation of the Nastran structural model for the subsequent construction of an aeroservoelastic numerical model for the design of a gust alleviation control system, following the modeling approach of Karpel and Moulin [5]. Three collocated grid points, rigidly interconnected in all directions except z, were placed at a location that represents the tip of the PZT actuator. The z displacement of the first point (z1) was constrained to be the average of those on the hinges of the flap and the tab. The z displacement of the second point (z2) was connected to z1 through a linear spring with the stiffness coefficient kv = 450.7 N/m that reflects the actuator properties given above. z2 was also constrained to z3 and the electric input by the MPC equation

z2 = z3 − xvv where v is the translation of a scalar point that was added to the model to represent the voltage input. With xv = 7.1 µm/Volt, a unit translation of the scalar point (representing v = 1 Volt) causes actuator displacements and forces that are consistent with the given piezoelectric characteristics and the dynamics of the surrounding structure [5]. Finally, z3 was connected to the tab rotation degree of freedom using a rigid element. The PZT actuator modeling is completed by loading the scalar displacement 6 v with a large fictitious mass of MH = 10 kg. Normal modes analysis with the resulting model yields a rigid body mode of tab deflection only. This mode is to be used as the control mode in the subsequent aeroservoelastic (ASE) analyses described below. The remaining frequencies and mode shapes are practically identical to those that would be obtained by adding the constraint v = 0, which represents the actual structure. These modes are slightly different than those obtained for the Nastran FRF analysis described above because the model now contains the tab mass and stiffness properties. The resulting tab natural C 6 S. Heinze and M. Karpel frequency is 29 Hz, significantly above the frequency range of the important aeroelastic activity shown below. The new Nastran modes were exported for subsequent stability and response analyses using ZAERO (see below). Doublet-Lattice aerodynamics [12] were used for computing the aerodynamic influence coefficient matrices for the Nastran FRF analyses. Aerodynamic panels were defined on the sections and splined to rigid body elements transferring the aerodynamic loads to the structure in the attachment points. On each of the sections, 4 spanwise and 10 chordwise aerodynamic panels were used. On the second outermost section, 11 chordwise panels were defined, where six were placed on the section, three on the flap and two on the tab. The total number of panels used for the half-span wing became thus 404. The number of panels in the chordwise direction was doubled, and it was found that the results only changed marginally. Therefore, the number of panels was considered sufficient.

ZAERO The Nastran normal modes with the detailed PZT actuator were used for generating the numerical model for ASE analysis and design using the ZAERO software package [13]. As discussed above, the first normal mode was used as a control mode that generates the aerodynamic and inertial forces due to the activation of the PZT device. Nine elastic modes, up to 37 Hz, were taken into account, which resulted in 18 structural states. The frequency- domain aerodynamic coefficient matrices were generated by the ZONA6 option of ZAERO based on the same paneling as used for the FRF analysis in Nastran. A column of gust loads was added to the aerodynamic matrices to represent the load distribution due to sinusoidal gusts normal to the wing. The minimum- state aerodynamic approximation technique [14] was used to approximate the resulting generalized aerodynamic matrices as rational functions of the Laplace variable s, such that a state-space time-domain ASE model could be generated with 4 aerodynamic lag states. The frequency bandwidth of the PZT response to voltage command is typically much larger that the aeroelastic frequency range of interest [1]. This would allow the modeling of the PZT actuator as a zero-order control law with no delay. However, the control-mode ASE modeling methodology [5] requires the actuator to be modeled as a 3rd-order transfer function for proper inclusion of the control-surface inertial and aerodynamic effects. Hence three states were used for modeling the actuator by a T (s) = ac a + bs + cs2 + s3 with a = 6.6155 · 109, b = 5.2861 · 106 and c = 2.8159 · 103, such that it has a frequency band of more than 300 Hz, which implies a very fast response with Piezoelectric Tab for Aeroelastic Control Applications C 7 practically no delays. The resulting 25-state ASE model was exported for control design using Matlab with the wing-tip acceleration, wing-tip displacement and aileron rotation angle serving as output parameters, and with the actuator input command serving as an input parameter. The ASE model was first used for open-loop flutter analysis using the g- method [15] of ZAERO. The purpose of the flutter analysis was to investigate the effects of the added flap stiffness (discussed above) on the open-loop stability characteristics. The variations of the aeroelastic frequency and the nondimensional damping versus air velocity are shown in Figure 3 for the free- floating flap and in Figure 4 for the stiffened flap with rotational stiffness.

5 Frequency [Hz] 0 0 5 10 15 20 25 30 35 40

0 Damping

−1 0 5 10 15 20 25 30 35 40 Velocity [m/s]

Figure 3: Frequency and damping variations vs. velocity, free-ﬂoating ﬂap.

As commonly done in presenting flutter results, all the presented damping values are negative when the system is stable. Flutter occurs when a damping branch becomes positive. The flap rotational frequency increase rapidly with velocity and crosses other modal frequencies. It is marked in the free-floating case of Figure 3 by “+” up to 5 m/s, and then by “◦” after crossing the first bending mode. Since the flap is underbalanced, it creates two flutter mechanisms when its frequency crosses that of the first and second bending modes, respectively. It starts in this case with f = 0 Hz at V = 0 m/s and crosses the first bending frequency at about f = 3 Hz and the second bending frequency at about C 8 S. Heinze and M. Karpel

5 Frequency [Hz] 0 0 5 10 15 20 25 30 35 40

0 Damping

−1 0 5 10 15 20 25 30 35 40 Velocity [m/s]

Figure 4: Frequency and damping variations vs. velocity, flap with rotational stiffness. f = 12 Hz. As a result, the first bending mode becomes unstable in a hump mode between V = 7 and 14 m/s (represented by the positive “+” at 10 m/s), and the second bending mode become unstable at V = 28 m/s (marked by squares). The rotationally stiffened flap rotational frequency (marked by “+” throughout Figure 4) starts with f = 7 Hz at V = 0 m/s. It interacts only with the second bending mode (marked by “”) to cause a flutter mode at 12 Hz at V = 26 m/s. Note that it was found that the flutter mechanisms for both flap configurations are hump modes, i.e. they are stabilized again for higher airspeeds. This was however not relevant for the present study, where only lower airspeeds were considered for the wind tunnel model. Hence, the stiffened case, which better represents the wind-tunnel model, is free from flutter in the test velocity range of up to 25 m/s, but its close flutter velocity caused difficulties in the control design, as discussed below.

Validation testing

Using the Nastran model, frequency response functions as those determined experimentally were computed. For comparison, the measured actuator strain and Piezoelectric Tab for Aeroelastic Control Applications C 9 accelerations were transformed to tab deflection and displacements, respectively. Magnitude and phase curves displacements in the wing tip accelerometer and flap deflections to angular deflections of the piezo electric tab are shown in Figures 5 and 6 for both simulation and experiment.

0.1 Experiment NASTRAN Experiment locked NASTRAN locked 0.05 Magnitude [m/rad] 0 0 5 10 15 20

180

Phase [deg] −90

−180 0 5 10 15 20 Frequency [Hz]

Figure 5: Normal displacement at location of accelerometer number 6.

Both magnitude and phase angle can be predicted well. For low frequencies, the predictions do not match the experimental data very well, which is expected since the accelerations depend on the square of the frequency and thus the signal- to-noise ratio for this frequency region is fairly low. It is clear that the numerical model is capable of reproducing experimental results in the considered frequency region. The concept of using the PZT actuator not directly for deflecting a control surface, but rather to amplify the effect by using the floating flap, was investigated. For very low frequencies, the effect of the tab deflection and the resulting flap deflection counteract each other, and there is practically no magnitude gain due to the free-floating flap. For higher frequencies, however, Figures 5 and 6 show that the magnitude using the locked flap is significantly lower than in the floating case with realistic stiffness and damping. Also, there is a phase difference of 180◦ between the locked and the floating configuration with rotational stiffness, which can be explained by the fact that in the floating case, the flap is deflecting in the opposite direction of the tab. As the excitation frequency C 10 S. Heinze and M. Karpel

6 Experiment floating NASTRAN floating (stiffened) 4

2 Magnitude [rad/rad] 0 0 5 10 15 20

300

200

100

Phase [deg] 0

−100 0 5 10 15 20 Frequency [Hz]

Figure 6: Flap deflection. exceeds the eigenfrequency of the flap mode, however, the flap starts to deflect in-phase with the tab, leading to a vanishing phase difference between the two cases. A numerical investigation of the floating flap properties was performed. As expected, the magnitude can be significantly increased when decreasing the stiffness of the flap rotational degree of freedom. An investigation of the rotary inertia shows that the bandwidth is reduced as the inertia increases. Therefore, to obtain the largest possible response, a lightweight flap with low stiffness is favorable. This however will lead to degradation of flutter properties, since low stiffness and the lack of mass balancing may result in control surface flutter. In the present case, a reasonable tradeoff was found.

Active control for gust alleviation

The purpose of this section is to demonstrate the applicability of the smart tab concept for the alleviation of the wing response to discrete (deterministic) gust excitation. It was demonstrated in the previous sections that the actual installation of the piezo tab in the wind tunnel, with some stiffness and damping in the flap hinge due to the electric cables, is free from flutter up to V = 26 m/s and has a reasonable authority on the flap rotation. However, the early attempts Piezoelectric Tab for Aeroelastic Control Applications C 11 to use this model for gust alleviation at 25 m/s showed that the proximity to the flap flutter speed of 26 m/s caused a severe ASE instability when a controller was added. Hence, it was decided to demonstrate the piezo tab capabilities with a free-floating flap that has no rotational stiffness and damping. It is assumed that it is possible to manufacture a floating flap with very low rotational stiffness, and that the flutter of this configuration at 7 to 14 m/s can be eliminated by some means such as flutter-suppression control laws, mass balancing or locking the flap in this velocity range. The FRFs of the wing-tip displacement and flap rotation to tab excitation, as calculated with the state-space ASE model, are shown in Figures 7 and 8 respectively. Plots are given in these figures for the flap with and without

0.2 Flap stiffness and damping No flap stiffness and damping 0.15 Experiment

0.1

0.05 Magnitude [m/rad] 0 0 2 4 6 8 10 12 14 16 18 20

200

100 0

−100 Phase [deg] −200 −300 0 2 4 6 8 10 12 14 16 18 20 Frequency [Hz]

Figure 7: FRF of normal wing-tip displacement to tab rotation command, ASE model. rotational stiffness and damping. The comparison of the plots with rotational stiffness to the floating cases in Figures 5 and 6 shows that the differences in the range of 0 to 11 Hz are insignificant. The responses in the range of 11 to 15 Hz also similar, but those obtained with the ASE model are larger than those calculated by Nastran by a factor of 2. The differences are due to differences in the actuator dynamics (stiffness and inertia) and maybe also due to the slightly different aerodynamic coefficients between Nastran and ZAERO. Since the system is close to flutter, slight changes in the structural and C 12 S. Heinze and M. Karpel

15 Flap stiffness and damping No flap stiffness and damping 10 Experiment

5 Magnitude [rad/rad] 0 0 2 4 6 8 10 12 14 16 18 20

200

100

0 Phase [deg]

−100 0 2 4 6 8 10 12 14 16 18 20 Frequency [Hz]

Figure 8: FRF of normal flap deflection to tab rotation command, ASE model. aerodynamic models can cause large response differences. In any case, since the frequency range of interest for gust response analysis when the system is stable is up to about 5 Hz, as discussed below, the ZAERO ASE model can be considered as well representing the wind-tunnel model. The no-stiffness plots in Figures 7 and 8 show a significantly larger control authority compared to the ASE cases with stiffness, with much smaller responses at 12 to 14 Hz due to the larger difference between the flap and the second bending frequencies. The larger control authority and the larger distance from the flutter velocity make the no-stiffness model a better candidate for demonstrating the piezo-tab gust-alleviation capabilities. The goal of the control design was to reduce the structural accelerations in response to discrete-gust excitation as much as possible without exceeding the maximal PZT voltage command of 100 Volts and without causing instability at the tunnel velocity of 25 m/s. The main purpose was the demonstration of the piezo tab capabilities with the simplest possible control law, which allows physical insight. The test case was the dynamic response to the uniform normal gust velocity profile

wg(t) =w ¯g(1 − cos(2πt/Lg))/2 where w¯g = 0.5 m/s is the maximal gust velocity and Lg = 0.33 s is the gust Piezoelectric Tab for Aeroelastic Control Applications C 13 length in terms of the time it passes a point on the wing. The gust length was chosen to be equal to the wave length of the first bending frequency (3Hz), which was checked to yield the maximal open-loop acceleration response. The first-order low-pass controller k T (s) = c T s + 1 with k = 280 and T = 0.56 that reads the wing-tip acceleration and commands the PZT actuator was found suitable for demonstrating the tab capabilities. The time histories of the open- and closed-loop wing-tip acceleration responses and the closed-loop voltage command are shown in Figures 9 and 10 respectively. It can be observed that the control activity adds a significant

5 100

2 4 80 3 60 2 40 1 20 0 0 −1 −20 −2 −3 −40

Wing tip Z acceleration [m/s ] Open−loop system

−4 Actuator voltage command [V] −60 Closed−loop system −5 −80 −0.5 0 0.5 1 1.5 2 −0.5 0 0.5 1 1.5 2 Time [s] Time [s]

Figure 9: Gust response, wing tip ver- Figure 10: Closed-loop voltage com- tical acceleration. mand. damping to the structural response and that the maximal acceleration response (in absolute values) is reduced from 4.1 m/s2 to 3.1 m/s2, a reduction of 25%, without exceeding the maximal capabilities of the PZT actuator. The fact that the control authority is large at 3 Hz (see Figure 7) is not just good luck. As for the maximal open-loop response, the authority peak is also associated with the first wing bending frequency. The time histories of the open- and closed-loop tab and flap deflections are shown in Figures 11 and 12 respectively. Figure 11 also includes the commanded tab angle, which is the voltage command of Figure 10 kinematically translated to tab angle. The small open loop tab deflection and the small difference between the commanded and actual closed-loop tab deflections are due to the actuator stiffness properties. It can be observed that even though the extreme tab deflection is about 2.6◦ , the flap reaches 4.5◦ , the effect of which on the response accelerations is very significant. This ratio agrees with the corresponding case in Figure 8 at 3 Hz, the first wing bending frequency. C 14 S. Heinze and M. Karpel

2 5 Open−loop system Open−loop system 4 Command Closed−loop system 1 Closed−loop system 3 2 0 1 0 −1 −1 Flap angle [deg] Tab angle [deg] −2 −2 −3 −3 −4 −0.5 0 0.5 1 1.5 2 −0.5 0 0.5 1 1.5 2 Time [s] Time [s]

Figure 11: Tab deﬂection. Figure 12: Flap deﬂection.

Conclusions

The main advantages of piezoelectric actuators are the relatively small size and a high bandwidth. For practical applications, however, these actuators are quite restricted due to small actuator stroke and fairly high energy consumption. In the present study, it was shown by both numerical investigations and experiments that the concept of using a piezoelectric actuator along with appropriate aerodynamic amplification is a convenient way of exploiting the actuator. In the present case, the actuator stroke restriction was compensated for by the free-floating flap concept, and yet some bandwith could be preserved. In the present case, the highest performance in terms of frequency response magnitude and bandwidth is obtained by reducing flap stiffness and inertia. Practical issues as well as flutter concerns, however, require some tradeoff. A numerical gust-response investigation demonstrated that simple control laws based on local measurements can be used to alleviate the structural vibrations due to dynamic gust excitations by considerable amounts, 25% in our case. The maximal flap deflection of 4.6◦ obtained with a tab deflection of −2.5◦ is a good demonstration of the aeroelastic leverage that amplifies the effects of the very limited PZT stroke. This research forms the basis for further investigation on the exploitation of the piezo-tab concept in the alleviation of other gust response parameters and associated design loads. The demonstrated control authority indicates that such tabs may also be used for flutter suppression and aircraft maneuver en- hancements, which requires a further research. Piezoelectric Tab for Aeroelastic Control Applications C 15

Acknowledgments

All activities were ﬁnanced by the European Union under the Fifth Framework Programme, through the project Active Aeroelastic Aircraft Structures, project number GRD-1-2001-40122. The work was technically supported by Boris Moulin of Technion, Dan Borglund and John Dun´er of the Royal Institute of Technology.

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Robust Flutter Analysis Considering Modeshape Variations

Sebastian Heinze and Dan Borglund Royal Institute of Technology SE-100 44 Stockholm, Sweden

Abstract This paper outlines different means to take modeshape variations into account in robust flutter analysis. Applying robust analysis for assessment of structural variations shows that large perturbations can lead to significant variations of the structural mode shapes. Therefore, the fixed modal base approach is no longer sufficient and can lead to incorrect flutter results. Different means of accounting for those variations are studied. Taylor expansions of different order for approximating the variations of the normal modes with respect to structural variations are investigated, as well as increased modal bases to account for perturbations of the vectors in the nominal base. Also, an iterative approach for updating the modal base is presented. The different approaches are applied to a test case, where mass variation of a wind tunnel wing is considered. The performance of the different approaches is evaluated and it is found that some of the developed methods can be efficiently used to improve robust flutter results.

Introduction

Any aircraft analysis model is subject to simplifications to some extent, leading to uncertainties in the nominal model. Therefore, analysis based on the numerical model generally yields errors in the aerodynamic loads, structural behavior, and critical speeds. Robust flutter analysis usually aims at computing a worst-case flutter speed considering these uncertainties, but can also be used to consider larger variations, such as fuel burn. In the recent years, so-called µ-analysis [1] from the control community has been applied to perform robust flutter analysis [2, 3]. Most recently, Borg- lund [4, 5] combined classical frequency-domain aeroelasticity with µ-analysis to formulate the µ-k method, being closely related to the p-k and g methods [6, 7].

D 1 D 2 S. Heinze and D. Borglund

In particular, a simple and efficient µ-k algorithm that takes advantage of data from a p-k or g method analysis is described in Ref. [8]. At present, the structural uncertainty descriptions available in the literature are based on the assumption of a fixed modal base in the analysis [2, 4, 9, 10]. However, the modeshape variations that result due to variations of structural properties (mass and stiffness distributions) may have a substantial impact on the aeroelastics because they directly influence the unsteady aerodynamic forces. In this study it will first be demonstrated that the assumption of a fixed modal base can lead to incorrect flutter results. After that, possible means to account for modeshape variations are presented and evaluated by considering a simple test case.

Robust ﬂutter analysis

The basis for robust ﬂutter analysis is the nominal model. In the Laplace domain, the nominal equation of motion for a ﬂexible structure with aerodynamic interaction is given by

2 2 2 2 F 0(p)v = p M 0 + (L /V )K0 − (ρL /2)Q0(p) v = 0 (1) where M 0 and K0 are the mass and stiffness matrices, respectively, and Q0(p) is the aerodynamic transfer matrix depending on the nondimensional Laplace variable p = g + ik, where g is the damping and k is the reduced frequency. The aerodynamic reference length for computation of the reduced frequency is L, and V and ρ are the airspeed and density, respectively. The vector of displacements in terms of n degrees of freedom of the finite element model is denoted v. Note that structural damping and the dependence on the Mach number have been omitted for simplicity, but can easily be included. The stated equation is a nonlinear eigenvalue problem that defines a set of eigenvalues p and corresponding eigenvectors v. Nominal stability is guaranteed when all eigenvalues of the nominal system have negative real parts. In addition to the nominal system, the uncertainty related to this system is defined. If the system is subject to linear parametric perturbations of the nominal matrices, it is convenient to define uncertain system matrices according to

M = M 0 + M L∆M M R (2)

K = K0 + KL∆K KR (3)

Q(p) = Q0(p) + QL(p)∆QQR (4) where ∆M , ∆K , and ∆Q are uncertainty matrices containing mass, stiﬀness and aerodynamic uncertainty parameters, respectively. Each of the uncertainty Robust Flutter Analysis Considering Modeshape Variations D 3

matrices may contain several independent uncertainty parameters δj, added to the nominal matrices by left and right scaling matrices according to Eqs. (2)-(4). These scaling matrices are chosen such that the total parametric uncertainty ∆ = diag(∆M , ∆K , ∆Q) belongs to a set S∆ deﬁned as

S∆ = {∆ : ∆ ∈ ∆ and σ¯(∆) ≤ 1} (5) where ∆ deﬁnes a block structure and σ¯(·) denotes the maximum singular value. Replacing the nominal matrices in (1) by the uncertain matrices Eqs. (2)-(4) yields an uncertain ﬂutter equation in the form

[F 0(p) + F L(p)∆F R]v = 0 (6) where

2 2 2 2 F L(p) = [p M L (L /V )KL (ρL /2)QL(p)] (7) T T T T F R = [M R KR QR] (8)

In the subsequent test case, the derivation of these matrices will be demonstrated −1 in a practical application. Pre-multiplying (6) by F RF 0 and deﬁning the −1 system matrix F (p) = −F RF 0 F L leads to the form

(I − F (p)∆)w = 0 (9) of the uncertain flutter equation, where I denotes the unitary matrix and w = F Rv is defined. This form is well-defined except at the nominal eigenvalues where F 0(p) is singular. Robust stability of the system is guaranteed when the system is nominally stable, and when the uncertainty cannot desta- bilize the system. When the flutter equation is posed in the form (9), the µ-k method [4, 5, 8] can be used to find the airspeed making a critical eigenvalue p = ikˆ possible for some ∆ ∈ S∆. The corresponding reduced frequency kˆ will be referred to as the robust flutter frequency. For problems with purely real uncertainties, available solution algorithms require a computational effort that grows exponentially with the size of the problem [11]. The problem can be solved more efficiently when some complex- valued variation is introduced. In the most simple case, a small aerodynamic perturbation affecting the aerodynamic loads on all lifting surfaces uniformly can be introduced, which is done in the test case. The aerodynamic uncertainty provides enough complex uncertainty to make the problem feasible, and affects the flutter results only slightly. D 4 S. Heinze and D. Borglund

Modeshape variations

Solving (1) is in general computationally expensive, since even simple aircraft structures may require a large number of degrees of freedom n, making the involved matrices very large. A commonly used approach is to perform modal projection of the problem, where only m structural eigenvectors are considered. It is assumed that the critical ﬂutter mode shape can be represented by a linear combination of those eigenmodes, giving

T Z(∆) (F 0 + F L∆F R)Z(∆)η = 0 (10) where the n × n system matrices are reduced to size m × m by multiplication of Z = [z1 z2 ··· zm] containing m eigenvectors zj, and η is the modal eigenvector such that v = Zη. Typically, m is several orders of magnitude smaller than n.

Fixed modal base

Using the common fixed-base approach, it is assumed that the modal base in (10) is independent of the structural variations, i.e. Z(∆) = Z(∆ = 0) = Z0. The computational effort required to solve the nominal flutter equation is reduced significantly by the modal projection. Regarding the uncertain part of the equation, the modal projection will also lead to a reduction of the size of ∆. The maximum block size for each uncertain parameter δj will be reduced from n to m [8]. Since the computational effort using µ analysis depends strongly on the size of the uncertainty description, the modal formulation is also beneficial from this point of view. As known from traditional flutter analysis, modal projection leads to accurate solutions as long as the chosen subset of eigenvectors in Z0 spans a modal subspace capable of representing the actual flutter modeshape. The flutter modeshape is typically a linear combination of some of the lower structural modeshapes, and thus the m lowest structural eigenmodes are chosen to build Z0. As the structural model is subject to uncertainty and variations, however, the fixed-base approach implies a potential problem. For large structural variations that perturb the structural modeshapes significantly, the nominal modal base may not represent the actual flutter modeshape accurately, leading to incorrect results in the flutter analysis. In the following sections, different approaches to account for modeshape variations are presented. Robust Flutter Analysis Considering Modeshape Variations D 5

Perturbed modal base One possibility to account for modeshape variations is to use a Taylor expansion to estimate the perturbed modal base according to

Z ≈ Z0 + ZL∆SZR (11) where Z0 is the unperturbed modal base, and ZL and ZR are computed to represent the modeshape variation as a function of the structural variations in ∆S = diag(∆M, ∆K). Higher-order terms of the Taylor expansion have been omitted but can be included as well. In general, however, this approach would become infeasible, since higher-order terms contain cross-derivatives of the modal base with respect to the uncertainty parameters that would increase the problem size significantly even for quite few uncertainty parameters. For computation of the first-order coefficients ZL and ZR, an approach as presented in Ref. [12] can be followed, where the derivative zi,j of each eigenvector zi with respect to each structural variation δj is computed. Given the derivatives of the mass and stiffness matrices with respect to the uncertainty parameters

∂M ∂K M j = and Kj = (12) ∂δj ∂δj the eigenvector derivative zi,j is computed by solving the linear system 2 2 2 K − ωi MMzi zi,j [Kj − (ωi )jM − ωi M j]zi T = − 1 T (13) zi M 0 2 zi M jzi 2 T 2 where ωi is the ith eigenfrequency and (ωi )j = zi (Kj − ωi M j)zi is the derivative of its square with respect to δj. Finally, the expansion in (11) is obtained as     δi 1 rZ     X  δi   1  Z = Z + [z z ... z ]     0 | 1,i 2,i{z m,i}  ..   ..  i=1 . . Z Li δi 1 | {z } | {z }

∆Si ZRi (14) where rZ is the number of uncertainty parameters aﬀecting Z. The matrices

ZL = [ZL1 ZL2 ... ZLrZ ] (15) ∆ = diag(∆ , ∆ , ..., ∆ ) (16) S S1 S2 SrZ Z = [ZT ZT ... ZT ]T (17) R R1 R2 RrZ D 6 S. Heinze and D. Borglund can then be assembled. Higher-order derivatives of the mode shapes can be derived accordingly. The total system can then be written

T (Z0 + ZL∆SZR) (F 0 + F L∆F R)(Z0 + ZL∆SZR)η = 0 (18)

Using linear fractional transformation (LFT) matrix operations [13], the uncertain ﬂutter equation can again be posed in the form

(I − Fˆ (p)∆)ˆ w ˆ = 0 (19) where

 T T  0 ZLF L ZLF 0ZL ˆ F (p) = 0 0 F RZL  − 0 0 0  T  ZLF 0Z0   T −1 T T T − F RZ0 (Z0 F 0Z0) [ZR Z0 F L Z0 F 0ZL] (20) ZR

T T T Note that the higher-order uncertainties ∆S ∆, ∆S ∆S, ∆∆S and ∆S ∆∆S resulting from (18) are transformed to a linear structured uncertainty block ˆ T ∆ = diag(∆S , ∆, ∆S) of larger size. Besides the increased size of the uncertainty block, another major drawback is that the favorable upper limit of ∆ being in the order of m is no longer valid. In this case, the size of the projected problem would exceed the size of the full-scale problem, making the modal projection meaningless. The reason for applying it is here to investigate if a linear approximation of the modeshape variation as such is meaningful. If it can be shown that (18) yields accurate results, a more eﬃcient formulation could possibly be developed.

Updated modal base Another approach to solve (10) is to apply an iterative solution algorithm, where the flutter equation T Zi (F 0 + F L∆F R)Ziη = 0 (21) with fixed Zi is solved to determine the worst-case flutter speed. Then, the corresponding worst-case mass and stiffness matrices are computed explicitly for the worst-case perturbation ∆Si , and an updated modal base Zi+1 is computed by solving the eigenvalue problem

2 (K(∆Si ) + ωi+1M(∆Si ))zi+1 = 0 (22) Robust Flutter Analysis Considering Modeshape Variations D 7

where K(∆Si ) and M(∆Si ) are the updated stiﬀness and mass matrices computed according to Eqs. (2)-(3), respectively. The iterations are terminated when the worst-case perturbation has converged according to

||∆Si+1 − ∆Si ||∞ < κ (23) where κ > 0 is a specified tolerance parameter. In the case study, the µ-k algorithm described in Ref. [8] was used to compute the robust flutter boundary based on (21). The resulting robust flutter speed and frequency were then used to formulate an optimization problem similar to the one posed in Ref. [11] for finding the corresponding worst case perturbation,

min σ¯(∆i) (24) ∆i

subject to det(I − ∆iF (ik,ˆ Zi)) = 0 where F (ik,ˆ Zi) is the system evaluated at the robust flutter speed with the robust flutter frequency kˆ. The minimax formulation in (24) is nonsmooth, but can be reformulated as a smooth optimization problem [14]. The resulting largest singular value of ∆i, here denoted σ¯(∆i), is the inverse of the lower bound of the corresponding µ-value [1] and can thus be used for judging the result of the optimization. In case of large differences between the upper-bound of µ (equal to 1 for the converged µ-k algorithm) and the computed lower bound, it is likely that the global minimum to (24) was not found and that the computed worst-case perturbation does not correspond to the worst-case flutter speed. Note that in general, the µ value can not be evaluated for any ∆ = Cr×r of dimension r, but rather has to be determined by upper and lower bounds. The Matlab µ-Toolbox [15] provides tools for computing these bounds and was used for evaluation of the upper bound in the present study. Due to the nonconvexity of the optimization problem, convergence to the global optimum can not be guaranteed, and thus the optimization was restarted several times from random initial values for higher reliability of the lower bound.

Increased modal base A simple way to account for a potentially insufficient modal base Z is to increase it by a number of vectors. The larger the number of linearly independent vectors in the modal base, the higher the possibility that the modal base can represent the actual flutter mode shape accurately. Since large modal bases require more computational effort, the number of eigenvectors cannot be increased arbitrarily, D 8 S. Heinze and D. Borglund but an efficient increase of the modal base is desirable. Several means to increase Z were investigated in this study.

Additional eigenvectors The most straightforward approach is to increase the modal base by additional structural eigenvectors. These eigenvectors establish a base with linearly independent vectors, which is advantageous since it assures that the projected matrices do not become singular due to the projection. The drawback, however, is that the nominal eigenvectors are not related to the structural variations of the system, and adding nominal eigenvectors does not necessarily capture perturbations of the mode shapes due to the uncertainty.

Eigenvector derivatives Another possibility to increase the modal base is to compute derivatives zi,j of eigenvectors in the current modal base with respect to structural uncertainties, and to use the derivatives as additional vectors in the modal base. Again, the derivatives can be computed as described above. These derivatives can be considered as vectors pointing in the direction of the modeshape perturbation due to given uncertainty parameters. Using these derivatives, the modal base is thus increased in the direction of the parametric uncertainty by writing

Zext = [ZZ∆] (25) where Z∆ contains derivatives of the nominal base. The vectors in Z∆ are however not necessarily linearly independent of the existing modal base Z, and they may be similar to each other as well. Besides increasing the problem size excessively, including linearly dependent vectors leads to an ill-conditioned problem. To ensure that only relevant vectors are included, the added vectors were made orthogonal to the existing modal base by using Gram-Schmidt orthogonalization to ﬁnd an increased modal base where

T zi zj = 0 ∀ i 6= j (26) Note that with this orthogonalization, the generalized mass and stiﬀness matrices are no longer diagonal matrices.

Case study

As a test case, the wind tunnel model described in detail in Ref. [16] is considered. The 1.2 m semi-span model consists of a composite plate that is mounted Robust Flutter Analysis Considering Modeshape Variations D 9 vertically in the low-speed wind tunnel L2000 at the Royal Institute of Tech- nology as shown in Figure 1. A beam ﬁnite element structural model and doublet-lattice aerodynamics were used for numerical analysis. The aerodynamic model is described in more detail in Ref. [4]. To demonstrate the diﬀerent approaches in a simple way, a variable concentrated mass was put on the leading edge of the wing, at about 40% of the wing span as shown in the Figure.

Mass balancing

Figure 1: Wind-tunnel model with leading edge mass balancing.

Uncertainty description As proposed in Ref. [4], linear uncertainties in the system matrices are formulated based on physical reasoning. From an analysis point of view, uncertainties are treated in the same way as variations, for example fuel level variations. Linear mass variations can for example be written in the form

XrM M = M 0 + δjwjM δj, δj ∈ [−1, 1] (27) j=1 where M 0 is the unperturbed mass matrix, M δj is the scaled matrix representing the perturbation in the mass matrix due to a variation δj, and wj > 0 is the perturbation bound, that may conveniently be included in M δj. The number of perturbations of the mass matrix is denoted rM . Note that M 0 is not D 10 S. Heinze and D. Borglund necessarily the nominal mass matrix of the unbalanced wing, since particularly mass balancing modeling requires a perturbed M 0 in order to obtain the stated boundaries on δj. In the present case, the mass balancing was varied from 0 to 1.0 kg. Note that the variation is rather signiﬁcant, since the wing weight is approximately 1.6 kg without mass balancing.

Minimum-size description By rearranging (27), the uncertain mass matrix can be written

XrM XrM M = M 0+ δjwjM δj = M 0+ M L,j∆jM R,j = M 0+M L∆M M R j=1 j=1 (28) where the matrices M L and M R can be chosen in diﬀerent ways. In order to obtain minimum-size uncertainty blocks ∆j, a singular value decomposition

T M δj = U jSjV j (29) can be performed, where Sj is a diagonal matrix with the same rank as the perturbation matrix M δj. For each variation parameter δj, the corresponding scaling matrices are then chosen as

M L,j = U jSj (30) T M R,j = wjV j (31) and the variation parameter is isolated to a minimum-size uncertainty block ∆j = Iδj, where I is a unitary matrix of the same size as Sj. Thus, the minimum size of ∆j is equal to the rank of the perturbation matrix M δj. If a modal formulation is used, the singular value decomposition is applied in the same manner to the projected matrices in order to reduce computational eﬀort and obtain minimum-size uncertainty blocks. Finally, the scaling matrices

M L = [M L,1 M L,2 ··· M L,rM ] (32) M = [M T M T ··· M T ]T (33) R R1 R,2 R,rM are assembled along with the uncertainty ∆M = diag(∆1, ∆2, ··· , ∆rM ). To increase the eﬃciency of the µ analysis, an aerodynamic perturbation was introduced. Expressions for aerodynamic uncertainty descriptions were derived in Ref. [4] and the uncertain aerodynamic matrix can be written Robust Flutter Analysis Considering Modeshape Variations D 11

Q = Q0 + δQwQQ0 (34) for the case of a linear perturbation of the entire nominal aerodynamic matrix Q0 with a complex uncertainty parameter δQ and a real uncertainty bound wQ. The pressure coeﬃcients were allowed to vary within 2% from their nominal values, where a uniform variation on the entire wing was assumed. Minimum- size descriptions for the aerodynamic uncertainty are determined in the same way as demonstrated for ∆M .

Results

For convenient comparison, the results of the robust flutter analysis are summarized in Figure 2. The Figure shows the lower bound flutter speed as a function of the maximum value of the variable mass. It was found that, due to its position at the leading edge, any mass balancing would increase the flutter speed. The expected solution in all cases was that the most critical perturbation is a zero mass, leading to a robust or worst-case flutter speed equal to the flutter speed of the clean wing. Therefore, a correct robust analysis should result in a worst-case flutter speed independent of the maximum possible mass balancing. The flutter speed without mass balancing and without aerodynamic uncertainty was found to be 13.9 m/s, which can be compared to the case with the largest variation, where a nominal mass balancing of 0.5 kg and a variation of ±0.5 kg result in a nominal flutter speed in the region of 24 m/s. When the aerodynamic uncertainty was added, the robust flutter speed was found to be 13.7 m/s. As a reference served an analysis of the full-scale system without modal projection. The full-scale analysis is computationally expensive and can only be performed for very small uncertainty descriptions, such as in the present case. Generally, however, this approach would be infeasible, and the results only serve as a reference for comparison of the different approaches. The performance of the different approaches is judged by the deviation from the full-scale results. The robust flutter speed predicted by the full-scale model is 13.7m /s, see Figure 2.

Fixed modal base is the most simple approach since it reduces the problem size significantly. This approximation was the most computationally efficient. For the case considered, using three eigenmodes in the modal base gives fairly accurate flutter results if no structural uncertainty is present. As the structural uncertainty increases, however, the modeshape variation due to structural variations leads to a flutter modeshape that cannot be represented by the first three D 12 S. Heinze and D. Borglund

13.8

13.6

13.4

13.2

12.8 Fixed base, 3 modes 12.6 Perturbed base, 3 modes Robust flutter speed [m/s] Iterative base, 3 modes 12.4 Increased base, 6 modes 12.2 Increased base, 3 modes + 3 derivatives Full scale 12 0 0.2 0.4 0.6 0.8 1 m [kg] max

Figure 2: Comparison of the robust flutter speed for the different approaches. eigenmodes any longer, leading to an incorrect flutter speed. For the maximum considered variation, the flutter speed is underpredicted by more than 1 m/s.

Modal base perturbation was performed using linear approximations of the modeshape variation. This was the most computationally expensive approach. As shown in Figure 2, the linear approximation reduces the deviation from the full scale case especially for smaller variations compared to the fixed-base case. As the structural variation increases, however, deviation increases. This is mainly due to the actual modeshape variation not being linear with respect to structural variations. This leads to incorrect modeshape predictions as the variation increases. An investigation of higher-order expansions of the modeshape with respect to structural perturbations was performed. Results for some representative eigenvector are shown in Figure 3. The figure shows, in this specific case, that the first-order estimation overpredicts the modeshape variation due to a fairly large mass balancing. It even deviates more from the true mode shape than the unperturbed modeshape does, implying that the first-order estimation could increase the error. Considering second and third-order terms improves the estimation, but this increase the size of the uncertainty description significantly, making the problem infeasible to solve. Robust Flutter Analysis Considering Modeshape Variations D 13

0.9 Perturbed 0.8

0.7

Third order estimate 0.6 First order estimate

0.5

0.4 Second order estimate

0.3 Unperturbed Normalized twist deformation 0.2

0.1

0 0 0.2 0.4 0.6 0.8 1 Normalized span coordinate

Figure 3: Perturbation of the 3rd eigenvector due to 0.5 kg mass balancing, and modeshape estimations using Taylor expansions of diﬀerent order.

Modal base iteration combines the small size of the uncertainty description from the fixed-base approach with the possibility to account for modeshape variations. The computational effort in this case was about the same as for the fixed base approximation, multiplied by the number of iterations needed. As shown in Figure 2, the flutter speed is predicted very close to the flutter speed from the full-scale case. The error is in the order of the deviation due to the modal projection as such. This error is always present since the projection ne- glects flutter modeshape contributions of higher modes, and it is visible even for zero structural variation. The worst possible perturbation obtained by solving the optimization problem (24) was δM = −1, corresponding to the zero mass balancing. The iteration converged after one step, since both the initial and the updated modal base resulted in the same worst-case perturbation and thus fulfilled the convergence criterion in (23), where a tolerance parameter of κ = 10−3 was chosen.

Increased modal base Adding three eigenvectors to the modal base increases the accuracy signiﬁcantly, along with a slight increase in computational time. The deviation from the full-scale results is very small in this case. There is however a slight trend in the graph indicating that there may be some deviation D 14 S. Heinze and D. Borglund for increasing variations. Adding orthogonalized eigenvector derivatives to an orthogonalized modal base yields very accurate results. In the case presented in Figure 2, ﬁrst-order derivatives were computed for the three nominal eigenvectors, leading to an additional set of 3 vectors. Note that there is no solution for the case mmax = 0 in Figure 2, since there exist no eigenvector derivatives for this value of the mass.

Conclusions

The objective of this study was to evaluate different approaches for taking modeshape variations into account in robust flutter analysis. In general, when dealing with small structural variations, the fixed modal base is sufficiently accurate. Even though structural uncertainties as such can have great impact on the flutter behavior, the impact due to modeshape variations appears to be a secondary effect. For large variations, however, the modeshape variation may lead to considerable errors in the prediction of the flutter speed. A first order Taylor expansion of the modal base is not very useful, since in cases with small perturbations, modeshape variations can be neglected, and in cases with larger perturbations, the linear approximation can be even less representative than the unperturbed modal base. Higher-order Taylor expansions for the modeshape variation, on the other hand, lead to a substantial increase of the size of the uncertainty description, making the problem infeasible to solve. Two useful methods for taking modeshape variation into account were provided. First, the modal base may be increased by deriving an appropriate set of additional vectors to be included in the modal base. In the most simple case, adding structural eigenvectors can improve the results, but there is no guarantee that these eigenvectors capture the effect of the structural variation. It was shown that the robust flutter results can be improved significantly by including modeshape derivatives with respect to the structural variation. Second, the iterative approach was found useful for capturing modeshape variations without the need of modeshape derivatives. The main advantage with this approach is that a small uncertainty description is conserved in each iteration. A drawback is that computational time increases as several iterations have to be performed. Further, it is essential that the global optimum to the underlying optimization problem is found if the iterations should converge.

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Assessment of Critical Fuel Conﬁgurations Using Robust Flutter Analysis

Sebastian Heinze Royal Institute of Technology SE-100 44 Stockholm, Sweden

Abstract An approach to assess critical fuel configurations using robust flutter analysis is presented. A realistic aircraft model is considered to demonstrate how an available finite element model can be adapted to easily apply robust flutter analysis with respect to structural variations, such as fuel level variations. The study shows that standard analysis tools can be used to efficiently generate the system data that is required to perform robust flutter analysis. The µ-k method is used to compute the worst-case flutter speed and the corresponding worst-case fuel configuration is found. The main advantage of the proposed approach is that µ analysis guarantees robustness with respect to all possible fuel configurations represented by the tank model.

Introduction

In order to certify new or modified aircraft, a large number of flutter analyses has to be performed to ensure that all configurations of the aircraft are free from flutter. All aerodynamic and structural variations have to be assessed, for example mass variation due to fuel burn. Combining all possible variations, the number of configurations increases rapidly. Consequently, only a subset of all configurations can realistically be treated, meaning that the most critical configuration may not be detected. In this paper, the possibility to utilize robust flutter analysis to ensure flutter stability for a range of different aircraft configurations is investigated. With this approach, a numerical model that depends on a set of bounded parameters, defining the different configurations, is developed, and a robust flutter analysis is performed to determine the flight condition where some of the possible configurations becomes critical. The µ-k method [1, 2, 3] for robust flutter solutions is used for this purpose. The µ-k method extends standard frequency

E 1 E 2 S. Heinze domain flutter analysis to take model uncertainty into account and allows for re-use of existing flutter models and results. In a previous study, the impact of a varying concentrated mass on the flutter speed of a simple wind tunnel model was successfully treated with this approach [4]. This study focused on the influence of structural modeshape variations on the robust flutter speed. The present study aims at demonstrating the same technique in the case of a high-fidelity aircraft model, where robust analysis is used to assess critical fuel configurations. In addition, procedures to account for modeshape perturbations due to structural variations are further assessed.

Aircraft Model

A generic subsonic aircraft with a wingspan of 17 m and an aspect ratio of 15.5 is considered to demonstrate the approach. The aircraft is a glider aircraft with geometric and structural properties that are similar to typical high-altitude long- endurance aircraft. Since the aircraft has been investigated in previous studies, a Nastran shell model of the structure was available [5, 6]. Figure 1 shows the ﬁnite element discretization of the aircraft structure.

Figure 1: Shell model of the aircraft.

The model consists of approximately 12000 grid points with a total of n = 70000 degrees of freedom. 23000 triangular and quadrilateral plate elements as well as some beam elements are used. Unlike the original glider aircraft, tanks were modeled in the wings and coupled to the structure using linear interpolation elements in Nastran. These elements couple inertial forces from the tank to the structure without contributing to the structural stiffness. Approximately 2000 doublet-lattice panels representing the lifting surfaces Assessment of Critical Fuel Configurations Using Robust Analysis E 3 were defined in Nastran and interpolated to the structure using thin plate splines. Control surfaces were modeled to allow interaction between elastic modes and control surface deflections. Since it turned out that the critical flutter mechanism is symmetric with respect to the aircraft symmetry plane, the rudder and antisymmetric aileron degrees of freedom were locked in order to reduce the complexity of the model.

Modeling of fuel variation

A realistic fuel tank geometry for the aircraft was defined as shown in Figure 2, with the tanks being symmetric with respect to the xz-plane. The most simple way to consider fuel variation in flutter analysis is to introduce a parameter that describes the fuel level and to perform flutter analyses for a set of different fuel levels. Using multiple parameters to model different fuel distributions as well would however increase the number of required flutter analyses significantly. Using an approach based on robust analysis, multiple parameters can be treated more efficiently. To demonstrate this, the tank was discretized as shown in Figure 2, where a case with 4 tank elements is shown. This configuration is referred to as Configuration A.

y T1 T3 x

Figure 2: Tank geometry and discretization, Conﬁguration A.

Each of the tank elements Ti was modeled as shown in Figure 3. Note that it is essential to describe the mass variation as a polynomial function of some parametric variation δ, in the most simple case as a linear function. To obtain linearity in the present case, the tank boundaries were simplified as indicated in the Figure. This reduces complexity and size of the problem, as will be discussed later on. Note that the simplified description may be improved by compensating for geometric approximations by modifying parameters such as the fuel density. In the present study, it was not considered necessary to compensate for the simplified geometry. Using the notation defined in Figure 3, the fuel level li E 4 S. Heinze

z simpliﬁed boundary tank boundary y x

li(δi)

si ci

Figure 3: Simpliﬁed tank element.

and the fuel mass mi within each element become

δ + 1 l = i h (1) i 2 i δ + 1 m = i M (2) i 2 i with Mi = ρf sicihi being the maximum fuel mass, where ρf is the fuel density and si, ci and hi are the tank element dimensions as shown in the Figure. The fuel level li in each of the tank elements Ti is represented by the parameter δi with −1 ≤ δi ≤ 1, where δi = 1 and δi = −1 correspond to a full and empty tank, respectively. Using the simpliﬁed model, the mass moment of inertia around the center of mass of the fuel can be written

m (δ ) I = i i (l2(δ ) + s2) (3) xx,i 12 i i i m (δ ) I = i i (c2 + l2(δ )) (4) yy,i 12 i i i m (δ ) I = i i (c2 + s2). (5) zz,i 12 i i Assessment of Critical Fuel Conﬁgurations Using Robust Analysis E 5

Since mass and fuel level depend linearly on δi, the mass moment of inertia 2 3 depends on δi, δi and δi . Note that both mass and mass moment of inertia will contribute linearly to elements in the mass matrix of the ﬁnite element model. Therefore, the mass matrix depends on the variation parameter to the third power as well. If fuel level and fuel mass depend on δ to some higher order, the order of the terms in the mass matrix will increase accordingly. Also note that there is no coupling between the diﬀerent δi.

Varying mass matrix

In order to apply robust analysis, the mass variation must be posed in the form of a varying mass matrix that depends on some vector δ containing variation parameters δi. As noted above, the mass matrix can be written

Xr 2 3 M(δ) = M 0 + δiM i1 + δi M i2 + δi M i3 (6) i=1 where r is the number of tank elements and M ij are matrices to be determined. The matrix denoted M 0 corresponds to the mass matrix where all δi equal zero, which means that all tank elements are half ﬁlled. There are several ways to determine the matrices M ij. For simple mass variations with few nonzero elements in M ij, these matrices can be derived by hand. Due to a more complex variation of the mass matrix in the present case, Nastran was used to compute multiple M(δ) for a set of δ. Based on (6), the matrices M ij were then determined by

1 8 1 M = − M + M − M − 2M (7) i1 2 δi=1 3 δi=0.5 6 δi=−1 0 1 1 M = M + M − M (8) i2 2 δi=1 2 δi=−1 0 8 1 M = M − M − M + 2M . (9) i3 δi=1 3 δi=0.5 3 δi=−1 0

Note that the variation of δi is performed with δj = 0 for all j 6= i. Thus, (3r + 1) Nastran evaluations have to be performed to obtain all M ij and M 0. Note that Nastran is used to assemble the structural matrices of the system and to reduce the matrices by applying boundary conditions and multipoint constraints, but not to perform any analysis. Both nominal and robust ﬂutter analysis were performed using Matlab and the Matlab µ-Analysis and Synthesis Toolbox [7]. E 6 S. Heinze

LFT form of mass matrix To comply with the notation used in previous work on robust ﬂutter analysis [1, 4], Equation (6) can be rewritten according to

2 3 M(δ) = M 0 + M L1∆M M R1 + M L2∆M M R2 + M L3∆M M R3 (10) where

M L1 = [M 11 M 21 ... M r1] (11)

∆M = diag(δ1I, δ2I, . . . , δrI) (12) T M R1 = [II ... I] (13) specify the linear perturbation of the mass matrix due to the variation, and M L2, M R2, M L3 and M R3 deﬁne quadratic and cubic contributions, with matrices formed accordingly. The size of ∆M becomes (n × r) × (n × r). In many practical applications, however, the rank of the perturbation matrices M ij is very low, making ∆M excessively large, which increases the required computational eﬀort in the robust analysis. In order to minimize the size of ∆M , a singular value decomposition (SVD) of the perturbation matrices can be performed according to

T M ij = U ijΣijV ij, (14) where Σij is a diagonal matrix of size rank(M ij) containing the singular values of M ij. The varying mass matrix can then be written

Σ Σ Σ Σ Σ2 Σ Σ Σ3 Σ M(δ) = M 0 + M L1∆1 M R1 + M L2 ∆2 M R2 + M L3 ∆3 M R3 (15) where

Σ M L1 = [U 11Σ11 U 21Σ21 ... U r1Σr1] (16) Σ ∆1 = diag(δ1IΣ11 , δ2IΣ21 , . . . , δrIΣr1 ) (17) Σ M R1 = [V 11 V 21 ... V r1] (18) define the decomposed form of the first order mass uncertainty, and the higher- Σ Σ Σ Σ order contributions M L2, M R2, M L3 and M R3 are formed accordingly. Σ Note that the uncertainty matrices ∆i consist of r blocks, but that the size of these blocks in general may be different for different i, depending on the rank of the corresponding perturbation matrix. Assessment of Critical Fuel Configurations Using Robust Analysis E 7

In order to take the varying mass into account in the robust ﬂutter analysis, it is useful to pose the variations in the form of linear fractional transformations (LFTs) [8]. Figure 4 shows the varying mass matrix M(δ), mapping structural accelerations a to the inertial forces fM , and the equivalent LFT.

Σ ∆M f M M(δ) a P M a fM

Figure 4: Equivalent LFT description for M.

The mapping can be interpreted as a variable matrix M(δ) from accelerations to inertial forces and is equivalent to the varying mass matrix. From (15), 0 the matrices ∆M and P M are derived

 Σ  0 0 0 0 0 0 M R3    1 0 0 0 0 0 0     0 1 0 0 0 0 0   Σ  P =  0 0 0 0 0 0 M  (19) M  R2   0 0 0 1 0 0 0   Σ   0 0 0 0 0 0 M R1  Σ Σ Σ 0 0 M L3 0 M L2 M L1 M 0 Σ Σ Σ Σ Σ Σ Σ ∆M = diag(∆3 , ∆3 , ∆3 , ∆2 , ∆2 , ∆1 ) (20) Σ 2 Σ 3 which shows that the higher-order variations ∆M and ∆M are replaced by a linear variation of larger size. This can be done regardless of the order of the polynomial in (15), and higher-order descriptions of Eqs. (1) and (2) will Σ simply increase the size of ∆M . The mapping from a to fM can now be written as

Σ fM = M(δ)a = Fu(P M , ∆M )a, (21)

Σ where Fu(P M , ∆M ) denotes the upper linear fractional transformation shown in Figure 4. This form of the variation can now be exploited for assembly of the total system to perform robust analysis. E 8 S. Heinze

Robust analysis

The uncertain ﬂutter equation accounting for parametric variations in the system matrices can in general be written as a nonlinear eigenvalue problem

2 2 2 2 M(δM )p + (L /V )K(δK ) − (ρL /2)Q(δQ, p) v = 0, (22) where p = g + ik is the complex eigenvalue with damping g and reduced frequency k, and v is the eigenvector. The reference length is L, and the airspeed and air density are denoted V and ρ, respectively. In general, variations δM , δK and δQ can be defined for the mass matrix M, the stiffness matrix K and the aerodynamic matrix Q, respectively. In the present case, there is no variation in the stiffness matrix K. For purely real variations, available solution algorithms may require a computational effort that grows exponentially with the size of the problem [9], making the problem infeasible to solve for practical applications. The problem can be solved much more efficiently if some complex-valued variation is introduced. This can be accomplished by either introducing some small non-physical variation [7], or as done in the present study, by introducing some uncertainty in the aerodynamic loads. Some complex-valued uncertainty according to

Q = Q0 + QL∆QQR (23) is deﬁned, where Q0 is the nominal aerodynamic matrix, QL = wQI contains a real scaling factor wQ > 0, ∆Q = δQI contains the complex-valued uncertainty parameter δQ, and QR = Q0. The uncertainty corresponds to a uniform perturbation of the aerodynamic forces with |δQ| ≤ 1, where wQ bounds the perturbation. The uncertain aerodynamic matrix can be written as an upper LFT Fu(P Q, ∆Q), where P Q is derived as   0 QR P Q =   (24) QL Q0 The uncertain ﬂutter equation is then written in the LFT form

0 2 2 2 2 Fu(P M , ∆M )p + (L /V )K − (ρL /2)Fu(P Q(p), ∆Q) v = 0 (25) which can be posed in the form

Fu(P (p), ∆)v = 0 (26) Assessment of Critical Fuel Conﬁgurations Using Robust Analysis E 9

Σ using simple LFT operations, with ∆ = diag(∆M , ∆Q). With this formulation of the uncertain ﬂutter equation, structured singular value (or µ) analysis can directly be applied to detect the ﬂight condition where some structured ∆ with ||∆||2 ≤ 1 enables a critical eigenvalue p = ik at some reduced frequency k [3].

Modal Reduction The system matrices in (22) are generally large, in the present test case about 70000 × 70000. In order to reduce the computational eﬀort, modal projection is used. A subset of modal coordinates is introduced according to

v = Zη, (27) where the columns of Z hold structural eigenvectors zi, and the modal coordinate ηi describes the participation of the ith structural eigenvector. By projection of the flutter equation onto the modal subspace, the size of the eigenvalue problem is reduced to the number of considered eigenmodes, m. This number has to be chosen sufficiently large to obtain accurate results, but is usually several orders of magnitude smaller than the number of the degrees of freedom. Be- sides reducing the size of the system matrices, modal projection may also reduce the size of the uncertainty description. In particular in cases where the rank of the perturbation matrix is larger than the number of considered modes, the modal projection will lead to a minimum block size of the uncertainty matrix ∆, reducing the computational effort of the µ analysis. A concern in robust analysis with varying structural properties is however that the modal base Z depends on these variations. In a previous study [4], the impact of modeshape variations was investigated, and different ways to account for it were proposed. A summary of these approaches is given below.

Increased modal base Modeshape variations can be accounted for by including modeshape derivatives with respect to the r variation parameters. The extended modal base can be written

Zext = [ZZ∆] (28)

Z∆ = [v11 v12 . . . v1r v21 v22 . . . v2r . . . vm1 vm2 . . . vmr] (29) ∂zi vij = , (30) ∂δj where the eigenvector derivatives vij are computed as described in Ref.[10]. The idea of extending the modal base is to include new vectors that point in the E 10 S. Heinze directions in which the modeshapes are perturbed. For example, Figures 5 and 6 show the 6th eigenmode and its derivative due to fuel level variation in tank element 1.

Figure 5: Shape of the 6th structural eigenmode.

Figure 6: Derivative of the 6th structural eigenmode.

Including all possible derivatives in Z∆, however, may lead to similar vectors in the extended modal base, leading to ill-conditioned projected matrices causing numerical problems. Therefore, only the most relevant vectors in Z∆ are included. To formulate a criterion for selecting additional vectors, the derivative vectors are ﬁrst normalized to equal norm, and the orthogonal component ⊥ vij of each additional vector vij with respect to the vectors in the modal base is computed using Gram-Schmidt orthogonalization [11] such that

T ⊥ ⊥ T ⊥ Z vij = 0 and (vij ) vij = γij. (31) This ensures that only the perpendicular component of any eigenvector derivative will be added to the existing modal base. As a criterion for selecting the ⊥ most relevant additional vector, the norm of vij is considered, and the vector with the largest norm is chosen. Note that it is essential to include one vector at a time and to repeat the orthogonalization and selection to ensure that the included vectors are orthogonal to each other and to the current modal base, and to prevent similar vectors from being included.

Iterative modal base With this approach, the resulting worst-case conﬁgura- tion from the robust analysis is used to compute a new modal base. The new modal base is then used for the modal projection in the analysis. This is done iteratively, until the resulting worst-case perturbation matches the current modal Assessment of Critical Fuel Conﬁgurations Using Robust Analysis E 11 base. Although appealing due to preserving a small size of the problem in each iteration, a drawback is that the worst-case perturbation is computed by solving a non-convex optimization problem, and the global worst-case perturbation is not necessarily found.

Results

To obtain sufficiently accurate results, the first 40 eigenmodes were selected to establish the reduced modal base. Since the critical flutter mechanism is symmetric with respect to the x-z plane, only symmetric modes were considered, thus reducing the modal base to 19 eigenvectors. Nominal p-k flutter analysis [12] of the case with empty tanks gave a flutter speed of u = 100 m/s at a frequency f = 9.4 Hz. Adding an aerodynamic uncertainty with a bound of wQ = 0.02, the robust flutter speed without mass variations was reduced to u = 99 m/s. The aerodynamic uncertainty was then kept at the same level throughout the analysis. Thus, proper robust analysis should yield worst-case flutter speeds of 99 m/s or below. The robust analysis for finding the worst-case flutter speed and the corresponding fuel configuration was performed for two different tank configuration with 4 tank elements. In addition to Configuration A shown in Figure 2, Configuration B shown in Figure 7 was also investigated.

Figure 7: Fuel tank geometry of Conﬁguration B.

The robust results for the different tank configurations are summarized in Table 1. The first row summarizes the nominal flutter speed unom for the different tank configurations, where all tanks are half-filled (δi = 0 for all tanks). The robust flutter speed using the original modal base is denoted urob, and δwc is the corresponding worst-case configuration found by the µ-analysis that was used to update the modal base for the iterative approach. Results from the iterative algorithm for updating the modal base are included as well, denoted ˆ uˆrob and δwc, respectively. E 12 S. Heinze

Conﬁg. A Conﬁg. B unom 143 m/s 109 m/s urob 105 m/s 102 m/s δwc [−1, −1, −1, −1] [−1, −1, −1, 0.22] uˆrob 96 m/s 97 m/s δˆwc [−1, −1, −1, −1] [−1, −1, −1, 0.22]

Table 1: Robust ﬂutter results.

Configuration A In this configuration, the tanks were distributed inside the wing. The maximum fuel weight of the entire tank was about 60 kg per wing. Note that the weight of the aircraft without fuel and payload was 380 kg. The resulting nominal analysis for ∆ = 0 provided a flutter speed of 143 m/s. Using robust analysis, the robust flutter speed was reduced significantly to 105 m/s, along with a worst-case δi = −1 for all tank elements, corresponding to an empty tank. This result, however, is not robust with respect to the flutter speed of the empty-tank configuration, which is known to be 99 m/s with the aerodynamic uncertainty. The reason for this was found to be an insufficient modal base. In order to improve the result, the modal base was adapted to account for modeshape variations due to structural variations. Using the iterative approach for computing the perturbed modal base provided correct results, and a robust flutter speed of 96 m/s was obtained. This is slightly conservative with respect to the known flutter speed of 99 m/s, the reason for this being the fact that the µ-k algorithm is based on an upper-bound estimation of the µ value to guarantee robustness.

Configuration B Here, a slender tank with a total fuel weight of 16 kg per half-wing was considered. For the tank shown in Figure 7, the nominal flutter speed was 109 m/s. Using the robust analysis, the flutter speed was reduced to 102 m/s with the original modal base, along with a worst-case perturbation corresponding to a fuel distribution where only the rear tank element is filled. Again, this result is not robust, the reason being an incorrect modal base. For this tank configuration, it appeared that there is a fuel configuration being worse than empty tanks. Further, it was found that the resulting worst case perturbation is dependent on the tools used for computing the µ value. In general, the µ value can not be evaluated exactly, and has to be estimated by upper and lower bounds [7]. Figure 8 shows the upper and lower bound of µ as a function of the reduced frequency k for some given airspeed for Configuration B. Assessment of Critical Fuel Configurations Using Robust Analysis E 13

1 rear tank 100% filled

all tanks empty upper bound µ

rear tank partially filled

lower bound

k k rob

Figure 8: Upper and lower bound of frequency-dependent µ value.

The lower bound corresponds to some ∆ found by optimization, and corresponds to the exact µ value if the global optimum has been found. Since this can not be guaranteed, however, the upper bound must be considered to guarantee robustness with respect to the uncertainty parameters. In µ-k flutter analysis, the upper-bound peak is considered, and robust stability for some given airspeed is guaranteed when the peak is below 1. An obvious problem appears when the upper and lower bound differ significantly. In the present case, the peaks appear at different frequencies. According to the iterative algorithm, the worst-case perturbation is to be computed at the reduced frequency krob of the upper-bound peak, leading to a perturbation which is not worst-case according to the lower bound.

A number of nominal flutter analyses were performed for different filling levels of the rear tank, and it was found that a full rear tank in fact leads to the lowest flutter speed at 98 m/s with aerodynamic uncertainty, thus confirming the prediction of the lower-bound graph. Despite this, the modal base was updated according to the iterative algorithm using the perturbation corresponding to the frequency of the upper-bound peak, and it was found that the perturbation is close enough to the worst-case perturbation, and that the modal base was perturbed sufficiently to yield robust results. The resulting robust flutter speed of 97 m/s is slightly conservative with respect to the flutter speed found by the nominal analysis with a filled rear tank at 98 m/s. E 14 S. Heinze

Increased modal base It was found that increasing the modal base by modeshape derivatives did not perturb the modal base suﬃciently to obtain correct results in the robust analysis. One of the main problems when including modeshape derivatives is the computation of modeshape derivatives is computationally expensive for a realistic aircraft model with a large number of degrees of freedom. In addition, it is not obvious which derivatives should be included in the modal base to adapt the modal base to the perturbed structure without making the projected system matrices ill-conditioned. Finally, it was also found that simply increasing the size of the modal base did not improve results very much. This may be obvious since the perturbation does typically not disturb the structural modeshapes in a way that can be captured by a moderate number of additional eigenmodes.

Computational effort The most significant computational effort is spent on solving the µ problem based on (26), and the solution time was found to depend cubically on the size of ∆. As discussed above, the minimum size of the structured uncertainty for each uncertainty parameter is either the number of modes, or the rank of the perturbation matrix. In the present case, the rank of the perturbation matrices was 12 or less in all considered cases, whereas the number of modes used in the analysis was 19. Increasing the number of modes by modeshape derivatives had therefore no significant impact on the computational effort needed for solving the µ problem. In the present case, one µ evaluation with 4 variation parameters took approximately ten seconds on a 2.5 GHz Pentium computer. Note that the computational effort generally depends on the size of the structural (real-valued) uncertainties, and that the size of the aerodynamic (complex-valued) uncertainty in the present study does not have any significant impact due to its simplicity. Depending on the algorithm used for bounding the µ peak, up to 100 evaluations may be required to solve the robust flutter problem for one mode.

Conclusions

It was shown that robust flutter analysis can be used for assessing the influence of fuel variations on the flutter boundary. The presented approach is more efficient than considering different configurations by hand, and guarantees robustness with respect to the flutter speed for all possible configurations. From investigations with different fuel tank geometries and discretizations, it was found that there may be worst-case configurations that cannot be captured when modeling the fuel level in the entire tank using one parameter only. Although being a secondary effect in terms of the impact on the flutter Assessment of Critical Fuel Configurations Using Robust Analysis E 15 speed, the variation of the structural modeshapes due to fuel variations can play a significant role. In the present case, the flutter speed was over-estimated when neglecting this effect. Including eigenmode derivatives with respect to the uncertainty for extending the modal base seems insufficient, and is computationally expensive when modeshape derivatives of complex aircraft structures have to be computed. Using an iterative base yields good results at less computational effort, but there is no guarantee for convergence of the iteration. The obtained solution can however be judged by considering the upper and lower bound of the obtained µ value to increase the reliability of the robust boundary. This study has also demonstrated that commercial analysis tools such as Nastran [6] can efficiently be used to generate the database needed to perform robust flutter analysis considering structural variations. Since these tools are frequently used in aircraft design and certification analysis, numerical models of the nominal configurations often exist, making robust analysis accessible with modest additional modeling effort. The µ-k algorithm is also efficient in terms of exploiting nominal flutter results, since the frequency of the nominal mode can be used as an initial guess when bounding the peak in the µ-k graph. When large structural variations are present, however, finding the critical peak is more difficult, since the peak location may deviate significantly from the nominal frequency due to the impact of the variations on the structural eigenfrequencies. In addition, the critical flutter mechanism may change, making it necessary to perform a robust flutter analysis of several modes.

Acknowledgments

This work was ﬁnancially supported by the Swedish Defence Materiel Adminis- tration (FMV) under contract 278294-LB664174 and the National Program for Aeronautics Research (NFFP).

References

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On the Uncertainty Modeling for External Stores Aerodynamics

Sebastian Heinze, Ulf Ringertz and Dan Borglund Royal Institute of Technology SE-100 44 Stockholm, Sweden

Abstract A wind tunnel model representing a generic delta wing configuration with external stores is considered for flutter investigations. Complex eigenvalues are estimated for the wind tunnel model for airspeeds up to the flutter limit, and compared to eigenvalues predicted by a numerical model. The impact of external stores mounted to the wing tip is investigated both experimentally and numerically, and besides the impact of the stores as such, the capability of the numerical model to account for increasing model complexity is investigated. An uncertainty approach based on robust flutter analysis is demonstrated to account for modeling imperfections. Uncertainty modeling issues and the reliability of uncertainty models are discussed. Provided that the present uncertainty mechanism can be determined, it is found that available uncertainty tools may be used to efficiently compute robust flutter boundaries.

Introduction

Flutter testing is rarely performed on full scale aircraft due to the high risk for structural damage and failure. Instead, flutter boundaries are computed based on a numerical model of the aircraft, and flight testing under less critical conditions can be performed to collect data for validation of the numerical model. Clearly, problems occur when the flight test data shows deviation from the numerically predicted data. Deviations are likely to occur for any aircraft configuration, since model imperfections and simplifications always lead to some uncertainty in the numerical model, and the question arises what impact these uncertainties have on the predicted flutter boundaries. In most cases, some parts of the numerical model, such as the mass properties of different components, are well known, whereas other elements, like the aerodynamic loads in some region of a wing with complex geometry, are known

F 1 F 2 S. Heinze, U. Ringertz and D. Borglund to be subject to uncertainty. Earlier studies [1, 2, 3] have shown how uncertainties can be introduced in the numerical model based on physical reasoning and known modeling difficulties. This study aims at demonstrating how uncertainties can be introduced in the numerical model, and how data points collected at subcritical conditions may be used to establish an uncertainty model that is capable of producing reliable flutter boundaries. In the present study, a wind tunnel model was designed to demonstrate and evaluate uncertainty modeling approaches. A fairly simple wing geometry and structure is chosen to minimize the errors introduced by modeling simplifications. The model complexity is then increased gradually, and analyses and experiments are performed for each configuration to detect modeling difficulties. An external store in the form of a wing tip missile is used for this purpose, and the impact on the flutter behavior is investigated, as well as the capability of flutter analysis tools to provide correct results for increasing model complexity. In cases where the numerical predictions deviate from the experimental results, robust analysis is applied and uncertainty modeling approaches are evaluated.

Experimental setup

The considered model is a generic delta wing conﬁguration as shown in Figure 1 with a semi-span of 0.88 m. The wing is mounted vertically on the ﬂoor in the

Figure 1: Delta wing mounted in the low-speed wind tunnel L2000 at KTH. Uncertainty Modeling for External Stores Aerodynamics F 3 low-speed wind tunnel L2000 at the Royal Institute of Technology (KTH). Due to the high risk involved in flutter testing, it was found convenient to design a low-cost wind tunnel model with a low level of complexity. The wing consists of a glass fiber plate with carbon fiber stiffeners to obtain the desired structural properties. The wing can easily be equipped with external stores, such as an underwing missile and a wing tip missile. In the present study, the impact of a wing tip store will be investigated.

Structural design The wind tunnel model is meant to represent a generic delta wing configuration of a fighter aircraft. Due to the length and velocity scales imposed by the wind tunnel environment, some properties such as the Mach number of the model will not represent the full scale model. The model was however scaled in terms of the structural properties with respect to the aeroelastic behavior. The objective was to obtain eigenfrequencies f such that the reduced frequencies k = 2πf · L/V are the same for both model and full scale aircraft, where L is a reference length, and the velocity V is chosen at a typical flight condition. In the present case, it was found that both the length scaling and the velocity scaling factors between model and full-scale aircraft are in the order of 10, making the frequency scaling factor equal to one. Therefore, structural eigenfrequencies of the model were chosen to be equal to the eigenfrequencies of a representative full-scale structure. Since flutter behavior typically is governed by the lower eigenfrequencies of a structure, the first bending and the first torsional eigenfrequencies were emphasized. To obtain specific eigenfrequencies as required by the aeroelastic scaling, a few design parameters were defined in the structural design. In general, both eigenfrequencies and eigenmode shapes can be controlled by the mass and stiffness distribution. In the present study, a fixed wing geometry was assumed, and a glass fiber composite plate was considered as a baseline structure. Carbon fiber composite spars were attached to the composite plate to control the structural properties of the wing. Two spars were mounted on both the upper and lower side of the wing, and the spar dimensions and positions were chosen to obtain the desired eigenfrequencies.

External store The main focus of the study is to investigate the impact of an external store on the ﬂutter behavior. A realistic wing tip missile was therefore manufactured and mounted to the wing tip as shown in Figure 2. The missile is mounted to a launcher beam that is assumed to be rigidly connected to the wing tip. Dif- F 4 S. Heinze, U. Ringertz and D. Borglund

Figure 2: Wing tip external store. ferent configurations were considered to identify the impact of different missile components on the flutter behavior of the wing. Figure 3 shows the five considered configurations. In order to reduce the effects on the flutter speed due to structural differences between the different configurations, mass balancing was used to account for different components once removed from the wind tunnel model. The mass balancing for the fins was located within the missile body, whereas the mass balancing for the missile was located within the launcher beam to maintain the aerodynamic shape of the wing tip region.

Numerical model

Numerical models of the delta wing were generated in both Nastran [4] and ZAERO [5]. The model is composed of shell elements for the wing, mass elements for the missile, and aerodynamic panels for both wing and missile. The missile assembly was assumed to be rigid.

Structural modeling The geometry of the structural elements of the wing plate was defined such that the location of the carbon fiber spars coincides with the location of structural grid points in order to allow for accurate modeling of the different material Uncertainty Modeling for External Stores Aerodynamics F 5

Conﬁguration 1: Full missile

Config. 2: Config. 3: Config. 4: Config 5: Canard fins only Rear fins only No fins No missile

Figure 3: Uncertain patches in wing tip region.

properties. Nastran CQUADR elements were selected for connecting the structural grid points of the wing plate. Figure 4 shows the discretization of the structural elements of the wing plate. Note that the location of the carbon fiber stiffeners is indicated in the Figure. The irregular discretization in the wing tip region is chosen to allow for accurate modeling of the launcher beam attachment. Material properties were derived in several steps, starting by the glass fiber plate without the carbon fiber spars. Vibration testing of the plate was performed to obtain material data by matching the measured eigenfrequencies to the eigenfrequencies predicted by the numerical model. The wing was freely supported to exclude effects due to imperfections of the boundary conditions once mounted to the wind tunnel floor. Material data from the manufacturer were used to determine material properties of the carbon fiber spars. Once attached to the glass fiber wing, another vibration test was performed to validate the material data. After that, the wing was mounted into the wind tunnel, and yet another vibration test was performed. The measured eigenfrequencies appeared to be somewhat lower than expected for a rigidly clamped wing, and it was found that introducing some flexibility in F 6 S. Heinze, U. Ringertz and D. Borglund

Structural elements Aerodynamic panels

Figure 4: Discretization of the numerical model. the numerical model of the clamping improved the frequency matching. The flexibility was chosen such that the first and second eigenfrequencies predicted by the numerical model were within 0.1 Hz from the measured eigenfrequencies from the vibration testing. The missile was assumed to be rigid, and only the mass properties of the missile were modeled. A grid point representing the degrees of freedom of the missile assembly was defined and attached to the wing tip. In the experimental setup, the missile launcher is rigidly connected to the wing at two chordwise locations, which was modeled by a rigid body element interconnecting the missile grid point to the respective grid points at the wing tip. Masses for the various components of the missile were then rigidly attached to that grid point. The structural damping was neglected in the structural model, and it was assumed that this simplification results in slightly conservative flutter predictions without any significant impact on the general flutter behavior. The numerical approaches presented in the following sections are based on a model without structural damping, but could easily be modified to include damping.

Aerodynamic panels The aerodynamic modeling was performed in both Nastran and ZAERO. Nas- tran CAERO1 elements were used to model aerodynamic panels for computation of doublet-lattice [6] aerodynamic loads. The wing surface was covered by 12 spanwise times 14 chordwise lifting surfaces equally distributed on the wing area. The launcher beam and missile body were modeled as flat panels parallel to the wing surface, since it was assumed that airloads generated by panels perpendicular to the wing surface can be neglected in the flutter analysis. De- pending on the configuration, between 200 (Config. 5) and 402 (Config. 1) aerodynamic panels were defined. In Figure 4, the Nastran aerodynamic panels Uncertainty Modeling for External Stores Aerodynamics F 7 for the full missile configuration are shown. The ZAERO model using CAERO7 elements utilizes the same discretization as the Nastran model for the aerodynamic panels for the wing, the missile launcher beam, and the missile fins. The missile body, however, was modeled using BODY7 elements. To connect the structure and the aerodynamic panels, a surface spline was defined for the wing plate using a SPLINE1 in both Nastran and ZAERO, and a beam spline was defined for connecting the missile panels to the grid point defining the missile motion, using SPLINE5 in Nastran and ATTACH in ZAERO.

Flutter results

In most applications, flutter testing of real aircraft is restricted to subcritical airspeeds due to the high risk involved in operating the aircraft in flutter conditions. As the airspeed approaches the flutter stability limit, the aeroelastic damping of the structure is reduced, leading to weakly damped oscillations once the wing is subject to external excitation. The oscillations correspond to complex eigenvalues in a linear stability analysis, where the real and imaginary part represent the damping and frequency of a particular mode, respectively. Rather than comparing the flutter speed from numerical predictions and experiments, eigenvalues can be compared instead to validate the numerical model without the need to operate the aircraft at the flutter limit. The wind tunnel model, however, was designed to operate at and even beyond the flutter limit, and eigenvalues representing both stable and unstable conditions can be measured and compared to numerical predictions.

Experimental results Experiments were performed at different airspeeds and for different missile configurations. To estimate eigenvalues from oscillations, the wind tunnel model was equipped with an accelerometer in the wing tip region to monitor the structural response to an external excitation. Figure 5 shows the measured oscillation for an airspeed below the flutter speed, where the oscillations decay due to the aeroelastic damping. The data was first collected for the most simple configuration without the missile attached to the wing. There is some beating in the wave during the first 2 seconds after the excitation, indicating that there are several modes present in the response. This can hardly be avoided when using impulse excitation as in the present case, since many different modes will be excited simultaneously. F 8 S. Heinze, U. Ringertz and D. Borglund Wing tip acceleration

0 5 10 15 20 Time [s]

Figure 5: Stable response for excitation at a subcritical airspeed.

The other modes are however significantly more damped and thus only one mode dominates the motion after the first few seconds. Increasing the airspeed beyond the flutter limit, the oscillations increase instead, as shown in Figure 6, indicating that one mode is unstable. In this case, there was no need for external excitation since any disturbance in the airflow can initiate the oscillations. Despite the wing being designed for large deflections, wind tunnel testing above the flutter speed is dangerous due to the steadily increasing amplitudes, and would eventually cause damage to the model. Experiments were primarily performed at subcritical airspeeds, and only a few experiments were run above the flutter speed in order to assure that the mode actually becomes unstable. From the measurements, complex experimental eigenvalues pexp = σexp + iωexp were estimated by identifying a state-space system based on the measured time series. Frequency fexp = ωexp/(2π) and damping defined as 2σexp/ωexp are then extracted from the eigenvalues. Fig- ure 7 shows the resulting damping and frequency versus airspeed. The flutter speed lies between 25 and 26 m/s, where the damping crosses zero, corresponding to a purely imaginary eigenvalue. Similar experiments were performed for all Uncertainty Modeling for External Stores Aerodynamics F 9 Wing tip acceleration

0 5 10 15 20 Time [s]

Figure 6: Increasing oscillation at a supercritical airspeed. conﬁgurations. Note that numerical predictions, that were obtained as described in the next Section, are included in the Figure.

Nominal analytical results The nominal analysis is performed by solving the flutter equation " # L 2 ρL2 M p2 + K − Q (p, M) η = 0 (1) 0 V 0 2 0 where M 0, K0, and Q0(p, M) are the nominal mass, stiffness, and aerodynamic matrices, respectively. The airspeed, air density and Mach number are denoted V , ρ, and M, respectively, and L is the reference length used to make the equation nondimensional. The flutter equation is a nonlinear eigenvalue problem with complex eigenvalues p = g + ik and vectors of modal coordinates η. The flutter stability limit is found as the real part of the eigenvalue becomes zero. The imaginary part of the eigenvalue is the reduced frequency k = ω · L/V , and the real part g = σ · L/V is a measure of the damping of the system. As for the eigenvalues estimated by the experimental time series, F 10 S. Heinze, U. Ringertz and D. Borglund the damping is then defined as 2g/k. Note that ω and σ used in the analysis correspond to ωexp and σexp from the experiment, respectively. In Figure 7, the numerical results are shown along with the measurements. The comparison shows that the flutter speed is predicted accurately by the nu-

5.6

5.4

5.2 Frequency [Hz] 5 19 20 21 22 23 24 25 26 27 28

0.01 Experiment Analysis (Nastran) 0 Analysis (ZAERO)

Damping −0.01

−0.02 19 20 21 22 23 24 25 26 27 28 Speed [m/s]

Figure 7: Comparison of numerical and experimental results for the configuration without the missile. merical model, although there is some deviation between the predicted and the measured damping. The Nastran model is slightly conservative at subcritical velocities but predicts the flutter speed very accurately, whereas the ZAERO model is less conservative and overpredicts the flutter speed slightly. The frequency trend is predicted reasonably well, apart from some measured data points close to the flutter limit, where a slight frequency drop between 24 and 25 m/s is visible. In general, however, the analysis captures the behavior of the experimental setup well.

Influence of external stores When attaching the missile to the wing, the flutter behavior is expected to change both in the analysis and in the experiment. Since mass balancing was Uncertainty Modeling for External Stores Aerodynamics F 11 used to account for the missile components, however, all configurations should have similar structural properties, and the main differences are expected to be due to different aerodynamic loads. The missile was attached to the wing in several steps, beginning with the missile body (Config. 4). Flutter results from this configuration are shown in Figure 8. The Figure shows that both the fre-

5.6 5.4 5.2 5 Frequency [Hz] 4.8 20 22 24 26 28 30 32

0.02

Experiment

Damping −0.02 Nastran ZAERO −0.04 20 22 24 26 28 30 32 Speed [m/s]

Figure 8: Flutter results with missile body attached. quency and the damping are captured well. The numerically predicted damping is slightly conservative with respect to the flutter margin, leading to a slightly underpredicted flutter speed. There is a constant offset in the frequency, which may be due to imperfections in the structural model rather than aerodynamic uncertainties. Nastran and ZAERO produce very similar results for this configuration. In the next step, the canard fins were attached to the model. The frequency and damping curves for this configuration are shown in Figure 9. Compared to the case without canard fins, the flutter speed was reduced by 1.5 m/s in the analysis, whereas the measured flutter speed was reduced by almost 3 m/s. This indicates that the analysis is not entirely capable of accounting for the canard fins. Due to the conservativeness of the damping predictions in the previous case, however, the analysis from both Nastran and ZAERO now fits F 12 S. Heinze, U. Ringertz and D. Borglund

5.6 5.4 5.2 5 Frequency [Hz] 4.8 20 22 24 26 28 30

0.02

0 Experiment

Damping −0.02 Nastran ZAERO −0.04 20 22 24 26 28 30 Speed [m/s]

Figure 9: Flutter results with missile body and canard fins. the measured data almost perfectly. The frequency is again predicted well with a slight constant offset. Next, the canard fins were removed again and the rear fins attached to the missile. Results from these investigations are shown in Figure 10. Attaching the rear fins apparently changes the flutter behavior of the wing significantly in the experiment. Although the analysis predicts a stabilizing effect of the rear fins, the difference between prediction and experiment increase significantly for increasing airspeed. In the experiment, the wing actually is free from flutter, since the critical mode turns into a hump mode with weak, yet negative damping in the considered velocity range. The hump mode is neither predicted by Nastran, nor by ZAERO. Large deviations can also be seen for the frequency of the critical mode. The constant offset from the earlier results is present for low airspeeds, but deviates more and more for increasing airspeeds. Finally, the complete configuration was considered by attaching the canard wings, with results as shown in Figure 11. As before, the canard wings lead to a destabilization of the wing. In the analysis, this can be seen by a reduction of the predicted flutter speed from 34 to 32 m/s for the Nastran model and from 36 to 34 m/s for ZAERO. In the experiment, the stable hump mode Uncertainty Modeling for External Stores Aerodynamics F 13

5.6 5.4 5.2

Frequency [Hz] 5 4.8 30 35 40 45

0.04 Experiment 0.02 Nastran ZAERO

Damping 0

−0.02 30 35 40 45 Speed [m/s]

Figure 10: Flutter results with missile body and rear fins. now becomes unstable. The differences between analysis and experiments due to the rear fins is still apparent, there is a deviating trend in both frequency and damping. Unlike the previous case, experiments were now restricted due to the flutter limit, making the deviation between analysis and experiment less apparent than in the case before, where experiments were continued 10 m/s above the predicted flutter speed. The investigations of the different configurations show that the numerical model is fairly accurate in most cases. As the rear fins are included, however, their influence cannot be fully captured by the nominal model. In the present case, the analysis predicts conservative results, with increasing conservativeness as the model becomes less reliable. In general, however, a deficient nominal model may also lead to non-conservative results. In the following Section, a robust approach will be presented where uncertainty modeling will be used to increase the reliability of the analysis, both in cases where the nominal model is fairly accurate, and where the nominal model is significantly less accurate. Due to the similar behavior of the Nastran and the ZAERO models, the robust approach is only based on the Nastran model. It is assumed that a robust approach based on the ZAERO model would produce similar results. F 14 S. Heinze, U. Ringertz and D. Borglund

5.6 5.4 5.2 5 Frequency [Hz] 4.8

29 30 31 32 33 34 35 36 37 38

0.02

Experiment

Damping −0.02 Nastran ZAERO −0.04 29 30 31 32 33 34 35 36 37 38 Speed [m/s]

Figure 11: Flutter results with complete missile.

Robust approach

Comparison between the experimental and predicted eigenvalues indicates that the numerical model captures the flutter behavior fairly well for the wing without any external store. As the missile is attached to the wing, however, the analysis is not sufficiently accurate to correctly predict the flutter limit. It is therefore assumed that the error is due to modeling imperfections that can be accounted for by introducing uncertainties in the numerical model. Instead of computing nominal eigenvalues from (1), the objective is to compute eigenvalue bounds by considering an uncertain flutter equation according to " # L 2 ρL2 M(δ)p2 + K(δ) − Q(p, M, δ) η = 0 (2) V 2 where the system matrices M(δ), K(δ) and Q(p, M, δ) now depend on a set δ of uncertainty parameters δi, defined such that the uncertain system matrices equal the nominal system matrices in (1) for δ = 0. By assembly of an uncertainty matrix ∆ containing the uncertainty parameters δi, (2) can be posed in a form where µ analysis can be applied to investigate possible solutions to Uncertainty Modeling for External Stores Aerodynamics F 15 the eigenvalue problem [2]. In previous studies [1, 2, 7], this approach has been used in order to find robust stability limits, where the µ value was computed to determine if the uncertainty can lead to a critical eigenvalue p = ik. In the present study, the focus is not only on the flutter boundary, but on a range of flight conditions, where robust analysis is used to determine the maximum variation of the eigenvalues at a given flight condition.

Uncertainty modeling

The uncertainty modeling based on uncertain system matrices as discussed above is convenient for introducing uncertainties based on physical reasoning. In many cases, some parts of the nominal model are known to contain uncertainty due to modeling difficulties, whereas other parts may be known to be accurately modeled. In the present case, the structural model has been tuned to fit experimental data from vibration testing experiments, and is considered accurate. Uncertainty is only introduced in the aerodynamic matrix, where a modeling approach as presented in Refs. [1, 2] is used to allow for uncertainties in selected aerodynamic panels only. The aerodynamic matrix Q0 is therefore partitioned into left and right partitions Q0(k, M) = L · R(k, M), where R(k, M) computes the pressure coefficients in each aerodynamic panel as a function of the modal coordinates, and L computes modal forces from the pressure coefficients. Aerodynamic uncertainty is introduced such that the pressure coefficients for a subset of aerodynamic panels i is allowed to vary according to cpi = cpi0(1 + wiδi), where cpi0 is the nominal pressure coefficient, and wi is a real valued uncertainty bound that is chosen such that |δi| ≤ 1. As shown in Ref. [2], the aerodynamic matrix can then be written

Q = Q0 + QL∆QR (3) where ∆ is a block diagonal matrix containing the uncertainty parameters δi, and QL and QR are scaling matrices based on R, L and wi. Due to the bounds on δi and the structure of ∆, the uncertainty matrix fulﬁlls ∆ ∈ S∆ with

S∆ = {∆ : ∆ ∈ ∆ and σ¯(∆) ≤ 1} (4) where ∆ denotes the block structure of the uncertainty matrix, and σ¯(·) denotes the maximum singular value. This uncertainty description is then inserted in the uncertain ﬂutter equation (2). F 16 S. Heinze, U. Ringertz and D. Borglund

µ-p analysis Previous studies [1, 2] have shown that linear fractional transformations [8] can be used to pose the uncertain ﬂutter equation in a form suitable for µ analysis [9]. The µ value is the inverse of the maximum singular value of ∆ ∈ ∆ that is required to make the uncertain ﬂutter equation singular for a given p, which due to (4) means that

µ(p) ≥ 1 ⇔ p is an eigenvalue of (2) for some ∆ ∈ S∆ (5)

µ(p) < 1 ⇔ p is not an eigenvalue of (2) for any ∆ ∈ S∆ (6) The µ analysis can be performed for any p at any flight condition. Comput- ing the µ value of some p that fulfills the nominal flutter equation (1) will lead to an infinite value of µ, since no uncertainty is required to make a nominal eigenvalue become an eigenvalue of the uncertain flutter equation. As the p value deviates from a nominal eigenvalue, the µ value will decrease and eventually reach 1, corresponding to a bound in the complex plane for the uncertain eigenvalue. Since the damping is more critical than the frequency in flutter analysis, focus will be on possible deviations in the real part of the eigenvalues due to the uncertainty. More details on the algorithm used for the µ-p analysis can be found in Ref. [10].

Uncertainty validation As uncertainty is introduced into the nominal system, not only the structure of the uncertainty, but also the bounds wi have to be specified. Naturally, the bounds have significant impact on the robust flutter results. A too small magnitude may lead to results not capable of capturing the behavior of the system, whereas a too large bound leads to flutter results being too conservative. In the present study, uncertainty validation will be used to estimate the uncertainty bound [10]. For a given uncertainty structure, the minimum bound that is required to include the experimental results into the range of the robust predictions is selected. Validation based on eigenvalues (p-validation) would for example result in a bound that assures that measured eigenvalues are a solution to the uncertain flutter equation (2). According to (5) and (6), the least possible bound for a set of measured eigenvalues pexp is found when µ(pexp) ≥ 1 for all considered pexp, and when µ(ˆpexp) = 1 for at least one measured eigenvalue pêxp, which in this case would be the decisive eigenvalue for the bound. In the present study, it was considered inconvenient to perform p-validation, since aerodynamic uncertainty was found to hardly allow for variation of the Uncertainty Modeling for External Stores Aerodynamics F 17 frequency of the eigenvalues. This leads to unrealistically high uncertainty levels in order to capture the slight frequency offset observed in the nominal results. Instead, validation with respect to the damping only was performed (g- validation), aiming at finding the bound that allows an eigenvalue of the uncertain flutter equation to capture the measured damping. Using this validation technique, the measured frequency of the critical mode is allowed to differ from the robust predictions. The goal of uncertainty validation is to establish a reliable uncertainty model by considering measured data points at airspeeds well below the flutter speed, and to use this model to predict the robust flutter boundary. This is particularly convenient in cases where flutter testing cannot be performed, and where only test data from airspeeds below the flutter speed is available.

Uncertain panels The uncertainty modeling allows for specification of different aerodynamic panels to be subject to uncertainty, and it was found convenient to select panels in the region where the model is most likely to contain uncertainty that has an impact on the flutter results. As the nominal and experimental results have shown, the missile assembly has significant impact on the flutter behavior. The entire wing tip region was therefore considered for the uncertainty modeling. Patches containing aerodynamic panels were defined as shown in Figure 12. Twenty patches were defined for the configuration with the full missile attached to the wing. Fewer patches were defined for the other configurations accordingly. In general, increasing the number of uncertain panels will increase the computational effort required for computation of the µ value. It is therefore desirable to only consider those panels having significant impact on the flutter behavior. One approach for finding these panels is to perform uncertainty validation considering one uncertain patch at a time, and to compute the bound required to validate a set of measured data points. Performing the validation for the full missile configuration with respect to the measured damping of data points in Figure 11 yields the results shown in Figure 13. In all cases, the decisive velocity was found to be the maximum airspeed of 37 m/s, where the measured damping deviates the most. The Figure shows that the first patch, located at the leading edge of the wing tip, would require an uncertainty bound of approximately 0.7, meaning that a variation of the pressure coefficients in this patch by 70% would be needed to explain the difference between the nominal analysis and the experimental data. It was found that selecting the most significant patches may establish an efficient and realistic uncertainty model. The Figure indicates the choice of the 7 most sensitive patches to obtain an uncertainty description of reasonable size. Note that the number of considered patches is somewhat arbi- F 18 S. Heinze, U. Ringertz and D. Borglund

18 17 12 14 13 8 11 20 7 10 6 9 19 5 4 3 2 1 16 15

Figure 12: Uncertain patches in the wing tip region.

trary, and the impact of this number on the robust ﬂutter bounds is discussed below.

Robust results The five presented configurations can be divided into two groups with significantly different behavior. Configurations 2, 4 and 5, where no rear fins are attached, show reasonable agreement between numerical and experimental results, whereas configurations 1 and 3, where the rear fins are attached, show a clear deviation between analysis and experiment. Robust results are therefore presented for two configurations with different behavior. Note that the robust analysis was performed based on the nominal model generated in Nastran. First, considering the most simple configuration without the missile, the validation was performed for one patch at a time, and it was found that the leading edge wing tip patch introduced the most significant uncertainty. Using this patch only, an uncertainty level of 20% was required to validate the uncertain model with respect to the measured damping, which was considered to be realistic. Figure 14 shows the measured and nominally predicted damping as a function of the velocity, as well as the robust boundaries based on the validated Uncertainty Modeling for External Stores Aerodynamics F 19

12 Wing tip Launcher Missile body Canard fins Rear fins

6 Required norm 4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Patch number

Figure 13: Required uncertainty norm for validation of experimental data using individual patches for the full missile configuration. model. In this case, the decisive test case for the validation bound was found to be 24 m/s, where the measured damping touches the robust boundary. All other experimental points are within the robust boundaries. Due to the conservative nature of the nominal model, the worst-case prediction is even more conservative. The validation would have produced similar results even if no data points would have been available at and above the flutter speed. In this case, the validation procedure would thus be particularly useful in cases where no flutter data points can be collected, since the bounds established at lower airspeeds can be used for robust analysis at higher speeds. In the second case, where the full missile configuration was considered, results are different. Figure 15 shows results for this configuration. Different uncertainty models have been used and validated with respect to the measured data points. In all cases, the decisive airspeed was found to be at 37 m/s, being the flutter speed in the experiment. The case with one patch only refers to patch 1 in Figure 12 being subject to uncertainty. As shown in Figure 13, an uncertainty bound of 0.70 was required for this configuration. The case with seven patches combines a selection of the seven most significant patches accord- F 20 S. Heinze, U. Ringertz and D. Borglund

0.05

0.04 Experiment Nominal analysis 0.03 1 uncertain patch worst−case damping 0.02

0.01 worst−case flutter speed

Damping −0.01 best−case flutter speed −0.02 best−case damping −0.03

−0.04

−0.05 19 20 21 22 23 24 25 26 27 28 Velocity [m/s]

Figure 14: Robust bounds for wing without missile. ing to Figure 13, leading to a required uncertainty bound of 0.24. Considering all 20 patches deﬁned in the wing tip region, the required uncertainty norm is further reduced to 0.18. The robust results seem to be fairly independent on the uncertainty structure once the uncertainty bound is adjusted based on model validation. In this case, the deviation between the nominal results and the experimentally determined values increases for increasing airspeed, which can not be repro- duced by the considered uncertainty. Instead, the required uncertainty bound has to be successively increased with increasing airspeed in order to capture the behavior. In this case, the deviation is likely caused by eﬀects that are not captured by the uncertainty description. Further investigations have to be performed to identify the source of the deviation, and it is possible that structural uncertainties must be considered as well.

Conclusions

This paper investigated the impact of an external store on the ﬂutter behavior of a delta wing conﬁguration. A wind tunnel model was used to provide Uncertainty Modeling for External Stores Aerodynamics F 21

0.05 Experiment worst−case 0.04 Nominal analysis damping 0.03 1 uncertain patch 7 uncertain patches 0.02 20 uncertain patches 0.01

Damping −0.01

−0.02 worst−case best−case flutter speed flutter speed −0.03 best−case damping −0.04

−0.05 26 28 30 32 34 36 38 Velocity [m/s]

Figure 15: Robust bounds for wing with full missile. experimental results for evaluation of nominal and robust flutter results. In order to identify modeling uncertainties, it was found convenient to model the missile in several steps, and to investigate how the flutter behavior changed gradually, both in the experiment and the analysis. It was found that the flutter behavior depends strongly on rather small missile details. This is mainly due to the location of the missile at the wing tip, where even small loads can have significant impact. Not only the flutter behavior as such, but also the performance and reliability of the numerical analysis may depend on rather small details, such as the missile rear fins. In many cases, when differences between analysis and testing are detected, finding the modeling error is rather difficult. In this study, the aerodynamic loads were assumed to be the most significant source of uncertainty, and uncertain parameters in the pressure coefficients were introduced to account for the modeling imperfections. The uncertainty models chosen in this study do not guarantee that the correct numerical model is covered by the uncertainty description. Instead, the uncertain patches were defined by engineering judgment, and the most sensitive patches selected. This selection was found to increase the computational efficiency without any significant difference in the robust flutter F 22 S. Heinze, U. Ringertz and D. Borglund boundaries. It was also found that if there is some deviation between nominal analysis and test data due to an inaccurate numerical model, it may be possible to find a model with an arbitrary aerodynamic uncertainty structure covering the data points, but it is unlikely that this model is capable of generating tight and reliable bounds unless the uncertainty model represents the true uncertainty mechanism. The performance of the robust approach was very different for different test cases. In some cases, the approach was found to successfully compute robust flutter bounds, whereas other cases demonstrated that this approach was less useful since the error apparently was caused by some phenomenon not captured by the uncertainty description. If unrealistically high levels of uncertainty are required, it is likely that the true uncertainty mechanism is not captured. The main challenge when using aerodynamic uncertainty to compute robust flutter bounds is thus to identify the main mechanism of the uncertainty and to develop an uncertainty description that can represent this behavior.

Acknowledgments

This work was ﬁnancially supported by the National Program for Aeronautics Research (NFFP).

References

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